This is kinda old news for most of the internet, but for those who don’t know, Adventure Time takes place in a post-apocalyptic setting as revealed by its creator.
According to the show, 1000 years ago the “Mushroom War” took place, and it is speculated that “Mushroom” refers to the mushroom clouds made in the explosion of nuclear bombs. Since then, life has begun anew, and mutation has caused quite a variety of new creatures.
In practically every episode, there is reference to our, now dead, civilization in the backgrounds. Broken buildings and cars and the like are usually partially buried in a lot of settings. In the opening of the show, dud nuclear missiles and broken pieces of tech are scattered in a wasteland and a gray and red (zombie-like) arm reaches out of a tree.
The earth is shown to have a chunk of it missing (It looks like North/South America is almost gone?). In the episode “The Lich”, the group travels over a body of water following the Lich to his lair, where he appears to use toxic waste (possibly from the nuclear war) to fuel his power. His hideout is in a destroyed subway station, and upon entering Finn and Jake are attacked by possessed modern-dressed skeletons, as if the people died in a bombing waiting for their train. On the surface, just outside, tanks and more dud bombs can be seen.
In the episode “Susan Strong”, the "Hyoomen" species lives in an underground abandoned city, where humans may have attempted to live after/during the war before dying off or mutating.
It is mentioned several times in the series that Finn is the last human, or that humans haven’t been seen for a long time. Marceline and the Ice King are speculated to be survivors of the the Mushroom War because of their ages.
Also one of the new episodes "I Remember You" and "Simon and Marcy", in season 4, shows a flashback of the world completely destroyed when Marceline was 7; being taken care of by Simon (The Ice King.). In "Simon and Marcy", destroyed cities can be seen, in inclusion to mutants, dud missles, and traffic, as if it was normal when the Armageddon fell.