Harold in the first three main games. He was only five years old when the bombs dropped and had to claw his way to top of the post-Nuclear Armageddon world, only to be knocked back down from being a successful merchant to a homeless drifter repeatedly. He also had the misfortune of being exposed to FEV during the Mariposa Expedition, which led to the creation of "Bob", a hunk of wood growing out of his head. Things go From Bad to Worse, once we cut to Fallout 3 Bob the hunk of wood, eventually overtook the rest of his body and rooted him into the ground, where he was stuck. As time went on, Bob grew into a small forest and Harold was worshiped as deity, by a small cult called the Treeminders, in-spite of his constant attempts to convince them otherwise, because they assume he's just "testing" them. By the time the player meets him in the Oasis that has grown around him, Harold is so sick of his miserable existence, he asks the player to Mercy Kill him. But the leader of the Treeminders and his wife want him to stay alive and to permanently keep Oasis to the cult or spread out Oasis, for the good of all the wasteland respectively, and the player has to choose between ending Harold's suffering and losing a potentially vital resource in rebuilding the world (his trees) or allowing it to continue, but at the benefit of the world as a whole. Though if you choose to help spread his "gift" across the wasteland, you can at least convince him of how much his "condition" can benefit the world as a whole and he'll find some solace in that.
Gob is a ghoul boy, that was enslaved and forced to work for Moriarty. He's apparently treated so badly by most people, he'll go behind his bosses back and risks getting beaten up to give you a discount, just for being nice to him.
If you decide to be a terrible person, you can make Amata's life hell growing up. Teaching bullies to mock her weight, killing her dad, destroying her home and causing her to be captured by the Enclave and shot to death for not revealing where Vault 101 is.
Bessie Lynn. While Michael Masters and Roy Phillips deserve to die for various reasons and revel in slaughtering the residents of Tenpenny Tower, Bessie is implied to not know what Roy's plan is, or at least may not approve of it, and her entire schtick is I Just Want to Be Loved, wanting to recapture the sense of self-respect and beauty she had as a human and following Roy because he makes her feel that way. If you choose to kill Roy and his followers and tell Bessie you're there to kill her, she barely fights back.
The Lone Wanderer his or herself, is likely this; You've spent your whole life living more or less happily in Vault 101 with your father, your life long best friend Amata, and several friendly neighbors (and a few unfriendly ones). Until one day, you're woken up by Amata and told a close friend of your family has been murdered, your father has ran off without saying a word, the Overseer is trying to have you killed for something, you didn't even know about, and you are forced to escape your own home and leave everyone else you've ever known, to wander the Capital Wasteland, fending off radiation, raiders, feral ghouls, super mutants, and many other things trying to kill, trying to find your father and hopefully some answers. Then after you've gone through hell and back to get back your father and your answers, you're forced to watch him perform a Heroic Sacrifice, while you can do nothing. Oh, and afterwards, you briefly return to your old home in Vault 101, to find that it has descended into anarchy and infighting following your departure, with many of its residence blaming you and your father for the current state of things, and even after you sort out this mess, many of them stillhold a grudge against you and you're forced to leave for good, by your best friend no less.
You could make an argument for pretty much every human the aliens abducted in the Mothership Zeta DLC being this, but a few stand out:
Though she doesn't really show it, Sally lived through and lost both her parents during The Great War, before being abducted along with her older sister and presumably only surviving family member, who is all but stated to have died sometime between when they were abducted and when the game takes place.
Paulson. While he may border on being a Jerk Ass Woobie, who could blame him, considering the aliens murdered his wife and son.
Possibly the biggest example on the ship is Elliot Tercorien. Like the rest of the team (with exception The Lone Wanderer and Somah) he is a Fish out of Temporal Water, but unlike them, who take that part of their situation in stride, he is devastated to learn of that the country that he and his fellow soldiers fought so valiantly to protect has been destroyed and the battles they fought in were rendered moot. While in the alien's cryogenics lab, he sees three of his former squad mates butchered on the operating tables, causing him to go on a brief Roaring Rampage of Revenge. Then afterwards, you and him find a couple more of his former squad mates still alive and frozen in the lab, but so badly injured from the alien's experiments, that they are guaranteed to die in short time, if they are unfrozen, leaving you and him with the tough choice of whether to give them a painless death by cutting off their oxygen or unfreezing them, so they can spend their last few moments fighting the aliens alongside the both of you, before they succumb to their wounds.
Veronica was forced to part ways with her old girlfriend, because the Mojave chapter of the Brotherhood of Steel needs everyone member to be in a relationship that actually allows them to produce a child, because they refuse to just recruit new members. She is also treated with very little respect and often kept away from the other members, because she questions the Brotherhood's self-destructive traditions, despite being the only one who's actually trying help them survive in the long run. If you finish her companion quest, I Could Make You Care, by persuading her to leave, she'll attempt to join the Followers of the Apocalypse, which results in a group of Brotherhood Paladins destroying a small outpost and its inhabitants. She not only blames herself for this but is so traumatized by what's happened that she refuses to enter the Brotherhood bunker with you anymore. Even not counting that, she still has to come to the realization that no matter what she does, her people will never change the ways, even if it destroys them in the end. No matter what you do, she's left with her spirit broken.
Boone. He's a Shell-Shocked Veteran of Bitter Springs who was also forced to Mercy Kill his wife (and by extension, his unborn daughter) in order to save her from becoming a Legion slave. After everything that he's been though, it's no surprise that he became The StoicCold Sniper.
Raul. Ever since the Great War, everyone that he cared about or loved died a horrible death with him being unable to do anything about it.
Cass' father left her at a young age. Then in her adulthood, the caravan company that she worked very hard to build up got wiped out just because she was caught in a conspiracy outside of her control. She's now broke and alone, sitting at a bar all day long. Until you come along.
Lily was living a peaceful life in a vault as a kindly grandmother before being taken by the Unity and transformed into a Nightkin. She still has memories of her grandchildren, who most likely died a long time ago. Even sadder when you encounter her for the first time and she confuses you for her probably long-dead grandson, Jimmy. She also doesn't get any happy endings. She either forgets her grandchildren entirely, remembers them just enough to think they're still alive and go on a futile search for them, or becomes completely Ax-Crazy as "Leo" takes over for good.
Arcade Gannon just wants to help the Mojave any way he can. He's troubled by his past, though it was really his father who was a soldier for the Enclave. You can help Arcade track down his old ex-Enclave friends who helped raise him as a boy and persuade them to fight the good fight in the end to make up for their past misdeeds, even though he's not in any way responsible for what happened. The worst part? If you do convince them to help fight with NCR or Legion at the battle of Hoover Dam, they will be rewarded by said group trying to arrest/kill them and Arcade and the other remnants will be become fugitives.
Jerry the Punk is a sensitive soul, who had the misfortune of being born into the Great Khans, a a tribe of proud warriors who (with the exception of Jack and Diane) give him no respect and even outright bully him, because he's not tough enough to pass their brutal initiation rituals. Luckily, the player has the option of offering him a better life with The Followers.
This seems to be a recurring trend with the NCR's First Recon snipers and not exclusive to just Boone, as mentioned above:
Sargent Bitter-Root's mother was junkie who either neglected him or tried to sell him for chem money, while his alcoholic father forced him to gun down NCR soldiers and civilians at three years old. He was also present at the infamous Bitter Springs Massacre, where it's strongly implied that he used the chaos to killed his parents for their years of abuse, an act his adopted father Major Dhatri believes he's traumatized by, but whether Bitter-Root himself secretly feels bad about this is left unclear.
Corporal Sterling used to be an NCR Ranger, until he was captured by The Legion, who tortured him so severely, his hand and feet were permanently damaged and he only escaped his captors, by throwing himself into The Colorado River to die, rather than let them have any more fun. Luckily, Sterling was saved by his fellow rangers and while he was to heavily injured to stay in the rangers, his skills with a rifle were still good enough to be kept around.
Corporal Betsy was raped by Cook-Cook an act that was so traumatic, she feels to need to hit on every woman that comes into camp to still feel in control and refuses to seek help, despite all of her squad mates wanting her to.
Dog is a violent glutton, but doesn't appear intellectually capable of actual malice. He's also in constant pain and has God verbally abusing him every waking moment, unless he mutilates himself enough to make the voices stop.
On the other hand, God is a mean-spirited Control Freak who's very frustrated that he can't get away from "the old man"/Elijah. In fact, if you complete Dead Money with God as the only personality, it's heavily implied that he's become nicer since he's finally free.
Siri used to have high hopes of becoming a doctor, only to see it all ripped away when the Legion invaded her home and made her a slave, leaving her as little more than a primitive medicine woman who can only craft tribal remedies due to the Legion's prohibition on science and advanced technology. Owe, and this being The Legion we're talking about, she has more than likely been raped while in the Legion's base.
Christine Royce is all but stated to be Veronica's former lover, who was forced to leave her, thanks to The Brotherhood's self-destructive traditions and Veronica's unwillingness to leave the only family she's ever known. When you find her, she has had her throat torn out and left trapped in a malfunctioning auto-doc, where she was alone in a dark cramped space, surrounded by the sound of malfunctioning surgical tools, leading to her developing Claustrophobia. Before that, she was trapped in the Big Empty's Y-17 Medical Facility after Elijah set her up as a distraction so he could escape. Once there, she was subjected to lobotomy via electrodes. The brain damage made it close to impossible for her to read or write, though she can still do mathematics just fine. Luckily, Ulysses broke her out before anything worse could happen. After the events of Dead Money, presuming she's still alive, she decides to willingly remain in The Sierra Madre, as it's personal guardian, both to keep people like Elijah out and the horrors in The Sierra Madre in, at the cost of sacrificing any chance she had at a happy life.
The Old World Blues DLC has Muggy, a neurotic miniature Securitron, built as an in-universe Take That! to Securitron creator Robert House, that Dr. 0 (who has a serious bone to pick with House) can take out his frustration on, largely by designing him to be obsessed with cleaning coffee mugs, but depriving him of access to any, so he's constantly begging for mugs. He's also not treated well by the other appliances in the sink. Plus in his Bad Ending, his tire gives out, just as he's about to reach a coffee mug, leaving him stuck with the mug he wants so badly just out of reach. But on the bright side, in the Golden Ending, he finds himself in Higgs Village, the resting place of the Think Tank's now abandoned homes, from when they were still human, where Muggy finds all the coffee mugs and other dishes, he could ever want to wash. He even gets a little personal revenge on Dr. 0, by leaving his dishes to gather dust, while he cleans everyone else's.
Randall Clark "The Father in the Caves", from the Honest Hearts DLC, was a former soldier who lost his family and home during The Great War and spent decades on the verge of committing suicide, while wracked with survivors guilt and feeling as though he had no purpose life. Whenever he finally finds something to live for, both in the form of a family of survivors that tried to help survive and a chance to start a new family with another women, it was yanked away from him. But he did eventually find peace, after becoming the guardian and mentor of a lost band of children, that years down the line, would become The Sorrows Tribe and would regard him as a god.
While all the Fallout protagonists have Woobie elements, this installment has the woobiest yet. Not only is there the usual tragedy of having to leave home into the devastated world but your spouse is dead, your infant child has been kidnapped, and you know firsthand what the world used to be like. As is addressed in-game in a conversation with the pre-war ghoul shopkeeper Daisy, the protagonist actually knows the difference. Comes off as an Iron Woobie fairly early given their response to the End of the World as We Know It, the death of their spouse, and kidnapping of their son is to become a wandering do-gooder who builds settlements. The issue is addressed in the interview with the Sole Survivor having the option of being devastated about the world, impressed with the way people are still trudging along, or sarcastic the biggest city left is a shanty-town in his/her favorite baseball stadium.
The Vault-Tec salesman. He comes off as rather corny and annoying in the prologue, but when you meet him as a ghoul two centuries later, well, it's hard not to feel sad for him. He's one of a handful of pre-war ghouls and might just be the loneliest man in the Commonwealth. If you try to reassure him that living in Goodneighbor isn't so bad, he retorts with: "If you kept hearing that 'keep your chin up' shit for two centuries, you might feel a little differently about it too." Giving him a home and a job in Sanctuary, on the other hand, greatly eases his depression.
Nick Valentine. Despite being a synth originally designed as an infiltrator, he is one of the nicest and most moral People in the Commonwealth. His artificial skin's also slowly tattering away over the years (exposing his internal components), at least two of the major power players in the region want him dead or enslaved just for existing, and he suffers severe existential angst/despair due to knowing that his memories, personality and morality base all come from the man his Neuralnet was based on to the point of feeling that nothing he does is truly "his own". Nick's had at least one shuddery moment thinking about what he'd be like if they'd uploaded someone like a Vault-Tec exec or condemned criminal instead. And to make matters worse, he's constantly having flashes of memories/events from his "former" life.
Paladin Danse, if you pursue his unique perk. The loyal and dutiful Brotherhood Paladin, one who believes in their mission to wipe the Institute and synths from the face of the Commonwealth, is revealed to be a synth himself, and he never even realized it! Elder Maxon will have none of it and orders him executed. You can save him from being killed, but he will be forced into exile. Everything he had lived for is completely turned on its head. What makes it even worse is that even when he knows the truth, he stubbornly keeps to his believes about synths, and demands that he be exterminated to set an example.
Cait. She grew up in an abusive family who locked her in a shed the first time she tried to run away, then broke her legs the second time. She thought that they kept her around because they really did love her deep down, only for her trust to be betrayed when they revealed their true intentions by selling her into slavery the second she turned 18. She was only able to escape her slavers by gradually stealing caps from them until she could buy her own freedom. She then got addicted to Psycho and found the Combat Zone where she was hired as a cage fighter. The Sole Survivor can potentially be the first person to ever treat her like a friend.
Oswald the Outrageous. He used to be a pre-war survivor who took shelter in the Kiddie Kingdom but became a Ghoul along with the other survivors. The radiation took a toll on the survivors and turned all of them feral over time, except him and his girlfriend Rachel. He dedicated his life to protect his ghoulified friends and the Kiddle Kingdom from the raiders with the use of irradiated gas, illusions and the ability to resurrect feral ghouls due to being a Glowing One. He spent 200 years guarding Kiddie Kingdom while waiting for Rachel to find a cure for going feral, unaware that Rachel herself was Driven to Suicide out of despair when she found out there was no cure. You can tell him to go help Rachel, tell him that there's no cure for going feral, give him Rachel's holotape, or outright attack him. Either way, if you know what happened to Rachel, no matter how you confront him, what you tell him, all his hopes are proven to be futile. It is hard not to feel sorry for both Oswald and the feral ghouls you mowed down to get to him.