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Tear Jerker: Star Trek: The Original Series
The ending of "Where No Man Has Gone Before", while the ship and crew survive intact (for the most part) it still feels like a downer ending. The fact that Kirk points out that they couldn't save Mitchell somehow makes it worse.
Spock's reaction to the Psi 2000 virus in "The Naked Time." The incurably stoic Vulcan breaks down in sobs. His explanation to Kirk of his regrets might also count, especially since Kirk is too pressed for time to listen.
Spock: Jim...when I feel friendship for you, I'm ashamed.
Spock says about Omicron Ceti III that for the first time in his life, he was happy. Awww, Spock!
The death of the Landru machine in "The Return of the Archons" was surprisingly sad in its own way. While Kirk was right in destroying it, hearing Landru cry out for its creator to save it as it exploded makes one wonder if perhaps the alien computer was sentient after all.
McCoy: I could have saved her! Do you know what you just did?
Spock: He knows, Doctor... he knows.
What makes it even more effective is that as Spock says this, the camera pans to Kirk, completely breaking down in tears. He does not come anywhere close to tears again until Spock's death in Star Trek II: The Wrath of Khan.
T'Pau: Live long and prosper, Spock. Spock: I shall do neither. I have killed my captain... and my friend.
This exchange is even more tearjerking when you consider the Fridge Brilliance factor of Spock's first sentence. He fully expected either to be disgraced and executed or to commit suicide (it's not certain which).
It's definitely suicide - the only death sentence left on Starfleet's books is for visiting Talos IV.
Or he was implying that he would essentially waste away from grief. From what's said about Vulcan mind-powers, it seems possible.
Spock's background potentially makes this even more heartbreaking. The guy has already gone through an amount of discrimination and bullying that has driven some people to villainy or to suicide, and his only response is to close himself off and work harder for acceptance. But when he realizes he's killed one of the only people to accept him (even if it's not really his fault), he breaks instantly, shutting down and wanting to die.
"The Doomsday Machine". Decker's anguished description of how he lost his entire crew. Also, Kirk begging Decker to come back and let them find another way besides a Heroic Sacrifice to destroy the creature. He doesn't listen.
Kirk: Matt, where's your crew?
Decker: On the third planet...
Kirk: There is no third planet!
Decker: (looks up at Kirk in horror and shame) Don't you think I know that? There was, but not anymore!
From "Journey to Babel", the scene between Amanda and Spock: recounting his painful childhood, her storming out in tears, his raised hand at the door, unable to do anything more...
Captain Kirk's descent into senility and paranoia in "The Deadly Years" is made more painful by the reactions of those around him, and especially Spock, who looks sadder and sadder as the episode progresses. When after the competency hearing Kirk calls him a traitor and adds he never wants to see him again, Spock leaves looking (by Vulcan standards) like he's about to cry.
"All Our Yesterdays". Kirk is able to get Spock and McCoy back to the present, but Zarabeth, the woman who helped them, the woman Spock has come to love, has to remain behind, condemned to live out the rest of her life alone in Beta Niobe's ice age past.
Spock: It happened, but that was five thousand years ago and she is dead now. Dead and buried long ago.
The ending of "The Paradise Syndrome". Kirk loses his unborn child and his wife within minutes of one another.
The deaths of so many Red Shirts. Yes, the death of Red Shirts was a Running Gag. But let's be honest, some of these deaths were very heartbreaking.
Poor Yeoman Thompson, who was turned into a mineral cube by Rojan and crushed in "By Any Other Name".
The young engineer Harper, who got vaporized by plasma flow activated by computer M-5 to restore its power supply. The worst part is how Daystrom tried to rationalize it as Harper simply "getting in the way."
While it's not mentioned afterward (figures) the two security officers who got beamed into open space while Sulu and Chekov were under the induced illusion that the ship was still in orbit of a planet in "And the Children Shall Lead". Poor guys...
The death of the man who was about to be married in "Balance of Terror," and his fiance's reaction at the end of the episode, is probably the saddest one in the series.
Part of this is that, Running Gag it may have become, to Kirk, the fact that these crewmen died on his watch is clearly a personal failing - he may have saved the ship, but his role as captain is to protect those men and women under his command, so to him, not coming home with everyone means he still failed.