Oxymoronic Being

"Aya: By the way, you're also a living human, aren't you? Why do you work in the Netherworld?

Youmu: Me? I've never thought of that. But I'm a dead human, too, so... hmm."

- Bohemian Archive in Japanese Red

Starfish Aliens and Eldritch Abominations are plenty strange but an Oxymoronic Being's mere existence is self-contradictory. To make one, take any two mutually exclusive characteristics; anything that somehow has both is an Oxymoronic Being.

And no, "honest politician" doesn't count.

Advertising

 * Starburst had been running advertisements with people like this for a while; for example, an albino lifeguard. One of them managed to be stunningly offensive by having the "contradictory" traits being Korean ancestry and Scottish nationality. It basically amounted to a painful national stereotype going off on a racist tirade against a kid.

Anime

 * In Naruto, Kisame's personal amount of chakra is great enough to draw comparisons to the Tailed Beasts, and thus he has given the oxymoronic nickname "The Tailed Beast Without a Tail ". To make this even more hilarious, in spite of his nickname
 * In an earlier volume of One Piece, Luffy and his first two crewmates Nami and Zoro, find themselves on an island inhabited by hybrid animals made up of a predator and prey animal. They find a rabbit-snake, pig-lion, and a raccoon-chicken.

Literature

 * The Phantom Tollbooth:
 * Milo meets an average-looking man who claims to be both "the smallest giant in the world" and "the tallest midget in the world". He's also the thinnest fat man and the fattest skinny guy.
 * There's also Canby, warden of the Island of Conclusions (you get there by jumping), who among other things is smart as can be and stupid as can be.
 * The triple demons of Compromise are described as "one fat, one thin, and a third exactly like the other two." (The illustrations show it as fat in part of its body and skinny in another part.)
 * Due to his being adopted, Captain Carrot Ironfoundersson of Discworld is a dwarf who stands at 6 feet 3 inches in his socks. The other dwarfs accept him as one of them because he is devoutly observant of their traditions and even tries to teach his girlfriend to speak Dwarfish. He's just a tall dwarf.
 * There's one story in an obscure German's childrens book about the inventor Scheibenkork who invented among other things the sphere with five vertices.

Myth and Legend

 * The mythical hippogriff is half-horse, half-griffin. However, since griffins hate horses and attack them on sight, it's hard to imagine how this could happen.
 * That was the point, originally. The Roman poet Virgil referred to the birth of hippogriffs alongside several other metaphors for things that couldn't happen in real life, in a "the lion shall lie down with the lamb" sort of sense.
 * The griffin is itself a compound creature, half lion and half eagle. Thus a hippogriff is a one-quarter eagle, which is $2.50 in gold. The study of zoology is full of surprises.

Tabletop Games

 * The Spelljammer setting in Dungeons and Dragons has enormous hamsters living between the worlds, appropriately known as giant space hamsters. There's also a lesser-known tiny variety, the miniature giant space hamster, which for most intents and purposes is merely a hamster. (They're best known for appearing in Baldur's Gate, in which one character has one as a pet. Well, maybe. He's not exactly sane enough to be a reliable source on that.)
 * Yu-Gi-Oh Card Game gives us the Anteatereatingant
 * The figures used in many tabeltop games are often called miniatures. They are still called this if they area foot and a half tall, such as the one representing Galactus in Hero Clix. Thus, they are large miniatures.

Toys

 * Action Masters are Transformers who are unable to transform after consuming Nucleon. Then there are Action Master Elites. who are Transformers that can't transform that can transform.

Video Games

 * The Horseless Headless Horsemann in Team Fortress 2. (Also, he has a head.)
 * Youmu Konpaku, from the Touhou games. She is a half human/half ghost swordswoman/gardener, with her ghost half existing as a formless white cloud separate from her physical body.

Web Comics
"Police Chief: So, the giant bear is...
 * Saturday Morning Breakfast Cereal: Handi-man is a quadraplegic superhero with the power to move like an able-bodied person.
 * This Order of the Stick comic.

Julia: Evil.

Police Chief: And the giant dwarf is...

Julia: An oxymoron."

Western Animation

 * Man-eating anteaters from Futurama.
 * Leela mentions returning from a vacation at Space Earth once.
 * Robot Chicken had a spoof of Attack of the 50-Foot Whatever movies called Attack of the Giant Midget. Said giant midget was about 6'5"... which is pretty tall, for a midget.
 * Sheep in The Big City has military characters with Punny Names that are usually oxymorons: General Specific, Private Public, Major Minor (who is a baby) and Corporal Ethereal (a hippy).
 * South Park has the Mexican Staring Frog of Southern Sri Lanka.
 * Another episode featured a group of extremist agnostics.
 * One villain from Sushi Pack, Paradoxter, enjoys things that are perfect paradoxes, such as sweet-and-sour sauce. His henchmen are jumbo shrimp (as in, man-sized), and in one episode he created animixes by splicing together two different animals, such as a sheep and wolf. One of his henchshrimp points out that he should call himself Oxymoron, since he's a Man-Ox, but Paradoxter does not appreciate this input.
 * Avatar: The Last Airbender:
 * The spider-fly, which commonly gets caught in its own web. Noted in the creator commentary as "the most paradoxical animal in the world of Avatar."
 * There is also the two-headed rat viper. This is used by the Air Benders as an analogy of consequences of anger: "Revenge is like a Two-Headed Rat Viper; while you watch your enemy go down, you are poisoned yourself."
 * When Sokka gives Toph a piece of the meteorite he used to make his sword he calls it "space earth", prompting Katara to question how there could be such a thing.

Real Life

 * The no-eyed big-eyed wolf spider. It is part of the big-eyed wolf spider genus, but has no eyes because it lives in a cave.
 * Similarly, there's a species of tailless bat that has a tail, so it's called the tailed tailless bat. Its scientific name is Anoura caudifera, which means "tail-bearing tailless one".
 * The Azorian Noctule, a bat that flies in daytime.
 * A feature of the Swedish language. It makes total sense to Swedes (and no one else) to describe something as "gigantically small".
 * The French word for "ridiculous" means "puny", the polar opposite of "insanely high".
 * In German, you can use "sau" (sod, sow, pig) to emphasize pretty much every adjective you like. Yes, it's perfectly reasonable calling food to taste "soddy good" and mean it as a compliment.
 * The Platypus. At the time of its discovery, most 'experts' concluded that the animal must be a forgery, because 'it is physically impossible for a creature to be both a mammal and a bird'. Actually a real creature, but few realized it at the time.
 * It's not a bird, but in fact a mammal, but it is an egg-laying mammal, which is also oxymoronic. Echidnas would also qualify for the same reason.
 * On the same note, in prehistoric times there were the marine reptiles, which were viviparous reptiles. Adding to they oxymoron, recent research shows that they (and dinosaurs) were also warm-blooded. In fact, turtles today are also warm-blooded, though not in the same way that birds and mammals are.
 * A species of titmouse bird is called Parus major, or the Great (meaning big) Tit. Awesome name by itself, but that's not all. One of the subspecies, the Japanese Tit, is taxonomically called Parus major minor, or Small Great Tit!
 * Ligers and Tigons would be considered an oxymoron as they inherit both the contradictory behaviors of their lion and tiger parents (lions hate water while tigers love it, lions are social while tigers are solitary, lions roar while tigers chuff, etc) often leading to a very confused creature.
 * Referenced by They Might Be Giants - when an interviewer called them 'the biggest indie act in the world', they said it was like being 'the world's tallest midget'.
 * Jumbo Shrimp.
 * Brachytrachelopan and Europasaurus, the small and short-necked sauropods.
 * Likewise, dwarf elephants.
 * The giant panda is both a carnivore and a herbivore. It's a carnivore because it pertains to the order Carnivora, but it's a herbivore since it eats only bamboo.
 * On that note, therizinosaurs, which were herbivorous theropods.
 * Some animals have a color name in their common name, but occur in colors other than that one, leading to a description that sounds a little like this when the animal is actually only one color. One example that's becoming increasingly easier to see is the "Red Green Iguana."

Rhetoric

 * The "Invisible Pink Unicorn": If it's always invisible, how do you know it's pink? Assuming it's even possible to know whether it exists.
 * Also the Invisible Red String of Fate, which is one of the things the Invisible Pink Unicorn references.

Web Original

 * According to Chuck Norris Facts, Chuck Norris holds every book in the Guinness Book of world Records, which means he's a planet, a pizza, a spider, and a distance.