However, as far as popular culture cares, martial arts are exclusively focused on fighting without any weapon or tool: they are about a fighter defeating an opponent solely through a trained usage of his body, overcoming natural traits as raw strength or size. This carries a deep inherent mystique, not only because it subverts the conceptions mentioned in the paragraph above, but also because someone being superior to others by training and knowledge is nearer to "magic" than something more obvious like armory or metallurgy. They also carry the implication that, as the body is the sole weapon, everybody can learn to use it, even you, because everybody has a body; meanwhile, weapons have been traditionally expensive and well regulated in history, leaving them much less accessible. Finally, the fact that some of these unarmed martial arts employ visually cool and/or flashy movements is another big point of attraction.
Exoticness is another important factor. The most famous martial arts in popular culture (Judo, Karate and Taekwondo) are all of Asian origin, which gives them the flair of the "mysterious orient" that fascinated the bourgeois western society of past centuries before globalization busted it. Religious and philosophic motifs play a big role, as Asian martial arts are usually associated to figures like ascetic Buddhist monks, elderly Chinese sages and honorable samurai, whose moral centers are popularly held much higher than their western homologues. In contrast, unarmed martial arts from the Old World like English boxing and European folk wrestling are still sanctimoniously portrayed often as the activities of brainless macho men who pursue violence as a way to live. (Even after the world fighting evolved away from this Oriental magic, other far countries like Thailand and Brazil have occupied part of their place for reasons related to the history of Mixed Martial Arts.)
Intuitively, portrayals of martial arts in popular culture are made to fit those conceptions, even if they dont always jive with reality. Judo will be a miraculous art in which a mouse can throw a cat around, karate will be a almost magic method for people to break bricks with a hand chop and a loud kiai, and kung fu will be a screaming superpower capable to beat dozens of opponents though flying kicks; and most importantly, all of them will be associated to a certain moral superiority, implying that their users have evolved past those gross tools specifically made to kill men called weapons.
This is often a mixed bag in popular culture, because weapons are still accepted as part of the mystique as long as they are associated to martial arts and/or artists, like the simple wooden staff (a metaphor for martial arts altogether: something completely mundane, like a cane or a stick, can turn into a lethal weapon in the right hands), the nunchaku (which looks cool and complex to use, therefore it must be more powerful than a simpler weapon, right?) or the legendary katana (as everybody knows Katanas Are Just Better). This contains some Truth in Television, as out of the more known and practiced martial arts in the world, judo is the only whose tradition faithfully excludes weapons: karate, taekwondo and many styles of kung fu do train with them often, including some of the ones mentioned above. However, they are usually relegated to a more zealous, traditional form of training in comparison to the good ol kicking. At the end of the day, unless context demans otherwise, barehandedness is better for narrative purposes, as it is seen as more sincere and wholesome: only you and your body against your opponent and his.
In real life, its generally understood that you dont want to fight unarmed if you can fight armed. Both ancient samurai and modern bar brawlers know that punching an opponent always hurts your fist at least a bit, but clubbing him with a stick doesnt, as the latter is not a part of your sensitive body. Also, obviously, weapons can always do things we are not naturally endowed to do, like cutting, piercing or shooting from a distance. The only intuitive scenario where youre expected to reject using a weapon is when you dont know how to use it and there is risk that it can be used against you or something worse. (For a better advice, just avoid fights at all its definitely much safer than anything you will read here.)
However, real life is again almost the opposite. The most battlefield-oriented martial arts have been historically based around grappling, as mankind invented shields and armor pretty fast, and as we said above, you gain little from kicking an armored opponent, except possibly a broken leg if your own armor doesnt protect it. Instead, grabbing the opponent and capitalizing on the reduced mobility of his armored body to control him was much more appreciated, as you gain a lot from wrestling him to the ground and placing him on a position where you can use your cool weapon to butcher him more easily. Grappling, especially on the ground and/or from the back, is less intuitive to humans than punching, so it becomes a form of fighting where training and knowledge mark a huge difference over natural toughness. To put a popular metaphor, even a lion is done if he falls in a pool with a shark. History of MMA first saw pure grapplers winning consistently over pure strikers, later saw experiments to find how to counter grappling, and ultimately saw people realizing they could do greater wonders with their striking if they added grappling to it or vice versa.
Grappling allows to control the damage inflicted, which is more adequate to contexts where life is not meant to be threatened, like sport competition or law enforcement. You can usually choose the amount of pressure you exert through a hold or the force you throw someone to the floor with in order to either completely break bones or merely produce pain, and in case you get caught yourself in a lock during a sport match, you can tap out before you suffer harm. However, only master strikers can control how much force they hit with, and its effects are typically a roll of dice; similarly, its not possible to "tap out" from a punch or a kick. You can also be hurt easily with your own strikes even if you connect them right, as the impact works in both ways, and thus affects both your opponent and the surface of contact you are giving out (a note: boxers use gloves to protect their hands, not to protect the other guys face).
Notably, this folder is an aspect of Artistic License Martial Arts that is slowly changing in modern times. With the rising popularity of MMA and the re-popularization of jacket grappling through Brazilian jiu-jitsu, which has brought a more specialized and educated kind of crowd, theres an increasing number of action films that are allowing themselves to include a lot of grappling.
This extends to disarming, as nothing shows the superiority of martial arts over weapons than a character suddenly going airborne and kicking the gun or blade off their surprised adversarys hands. In real life, this would be downright suicidal, not only because charging in a wide way would give your opponent a clearer target, but also because you could easily hurt your limb against the blade or making the gun fire accidentally with the impact. Disarming is a field that needs grappling to both impede the opponent to use the weapon and rip it from his grasp. Even so, however, it is always a last resort trick, which martial artists are explicitly taught not to try if the risk of getting hurt is not worthy.
This is possibly the most necessary artistic license in martial arts, because in audiovisual media, the crowd needs obviously to see the attack. Thus, in TV punches will be the widest and most telegraphed possible, kicks will be jumping spinning 720º corkscrew extravaganzas, and submission holds will be loose and hammy enough to give at least a hint of whats happening (see Striking vs. Grappling above), so the boys on the back row can see. Even a battle explicitly finished by catching an enemy off guard will usually receive a replay or an explanation to show how exactly the baddie was tricked. After all, people dont like to feel like fools, even if landing a correct technique is often about making a fool of the opponent.
Even the most acrobatic martial arts around are susceptible to this. Taekwondo often gives more points the more spectacular a technique is, while Capoeira has dance and gymnastics elements, so they are both naturally huge in the theatrical aspect. However, the former advises against doing such things in an environment that does not reward them with points, and the latter teaches to relegate fancy somersaults to friendly sparrings and to keep always a hand or foot on the ground.
Another point that pits pragmatism against practicality in media is the fighting stance. In competitive context, fighters adopt a very precise body stance meant to get them ready to both fire and block attacks, which goes adjusted to their ruleset (i.e., boxers will try to protect his head and upper body with his arms, kickboxers will open it a bit to prevent high kicks, etc.). In TV? Not so much. They know the audience rewards cool visuals, not people turtling up and measuring every jab, so they favor guys throwing wild haymakers and doing all sorts of complex stances and arm weaves. This is a bit of Truth in Television, as the least competitive martial arts do include some ceremonial stances that arent meant to be protective in an all-out brawl. Of course, movies dont care about the difference.
Footwork, or the way in which the fighter moves to outmaneuver his opponent and avoid being outmaneuvered himself, also goes included. In TV, again, not so much: people running around and leaping wall to wall is way more fun, and planting themselves down in a patch of terrain and exchanging hits like a boss on it has some badass component. Anime particularly loves having a character with Super Speed do a head-first charge to his opponent, freezing in front of them in a sort of "Im here" face to face moment, and only then breaking into strikes (in reality, this would probably meet a punch or a knee right on the nose). Then again, the world of martial arts is diverse enough to have some truth behind theatrical footwork; in Muay Thai, for instance, its traditionally approved to plod forward and engaging in stationary brawls as a sign of bravery.
Martial arts of the "soft" kind often use redirection and blending to throw the opponent down using their own momentum, but how the latter is created is a much more complex question than simply grabbing a big guy and expecting him to fall easily. An enemy charging wildly towards the hero is obviously vulnerable to a well placed trip or throw because he is generating a lot of momentum that can be redirected, but this kind of scenario doesnt happen easily outside of football or bullfighting. In fact, as we humans are vertical animals with two legs, a big part of our automated brainpower is dedicated to keep and regain balance, so jeopardizing it deliberately in a fighting context is simply not intuitive for us. As a consequence, most wrestling styles contain techniques designed to move the adversary to a stance where their kinetic energy can be exploited; expecting him to put himself on it is not enough to achieve it, especially if he knows how to avoid or counter it, which turns the match into a strength vs. strength contest. This is why most wrestlers and judokas need to be serious strongmen aside from sneaky technicians. Technique beats strength, but technique and strength beats technique alone. At the end of the day, you could say that using the opponents strength against him is viable, only that it should be added "...but you need to be strong as well to pull it, boyo."
Now about size, it is a mixed bag. Kung fu flicks love to put the lithe hero against some ugly giant and have him kicking his ass in a fight to underline his superior artfulness. But how realistic is this? Well, overcoming a size disadvantage by sheer technique is possible, if only situationally nuanced, but it becomes increasingly difficult the wider the size/strength difference is, and it turns even worse if the difference in technique is not wide enough towards the other side. As we said above, technique and strength beats technique alone. Even although MMA and kickboxing story have a few instances where gigantic guys were beaten by comparatively diminutive opponents, this not always happened by finesse alone, but by other additional advantages (for instance, many of those giant fighters had actually sub-par physical skills and/or couldnt adapt to the sports complex fighting game). The truth is that weight classes exist in combat sports for a reason. If your opponent is much bigger and stronger, you are in trouble, and if he is any decent on what you are both competing at, you are in a serious, serious trouble.
"The bigger the opponent is, the easier it is to take him down" is another difficult point. What determines the feasibility of a throw is the gravity center: a guy with a high gravity center will be easier to send tumbling down than one with a low one. Going by that rule, yes, a taller guy will be usually easier to unbalance, as his gravity center will be naturally higher than yours, especially if he has lanky legs or is not very flexible. However, a taller guy who has a low gravity center will be harder to unbalance, not easier. Sumo Wrestling is a great example, as although its wrestlers are usually huge, their training is designed to give them powerful legs in which their abundant weight can rest low. After all, its well known that you cannot easily overturn a pyramid, whose gravity center is on its very base.
That said, successfully fending off several opponents is not absolutely impossible. If you dive deep enough on YouTube, you can find (hopefully real) vids of petite female martial artists wrecking several surprised aggressors, hotheaded Russian boxers dropping multiple thugs in wild road brawls, and possibly European riot force cops caning people left and right. There are also stories of ancient judokas, old school professional wrestlers and MMA fighters getting in quarrels against ridiculous numbers of opponents and coming out alive to tell them. In a memorable anecdote, judoka Yoshiaki Yamashita apparently took on eighteen thugs by following the Spartan method in Thermopylae: he placed himself on top of a narrow ladder and forced them to come up to him one on one, where he could grab and toss them down individually with the safety that he would not be surrounded or outmaneuvered. Intuitively, this is often the vital element in those heroic brawls with happy ending: when the lone ranger wins, it is because he managed to turn a 10 on 1 in a series of 1 on 1, not because he could take them all at once.
Have in consideration this naturally requires an environment advantage, serious fighting instincts, being batshit insane and, last but not least, having enormous amounts of luck. Without just one of these, party is over. And even with all of them, it doesn't guarantee party keeps on. So, as we say, if an entire mob comes for you, its wiser and safer to run.