I don't have much evidence, as I don't yet have the tools to gather all the evidence, but I will incorporate several things from the show itself
Now, before I start spouting shit that 70% of you wont even read, adventure time is actually set in a post apocalyptic earth after a major nuclear war, with fin being (supposedly) the last human alive
My very first piece of evidence is the episode "heat signature", jake subtly makes a reference to something called "the great mushroom war". As you all know, an atomic bomb, or any nuclear weapon, makes a big mushroom cloud when it goes off.
Another example is in the episode where the business men wash ashore in huge iceburgs. First of all, they were all dressed up like a regular modern day business man would be, however their skin was tinted green, and they had sores and whelps all over them. Sounds an awful lot like a zombie, huh?
Remember the episode where the lynch gets free? well, after they traverse across the sea, for a few frames, the background is a huge city, but completely destroyed, with broken sky scrapers and busted up cars, and the "well of power" the lynch flees to is actually an old subway station
Finally, and the most recent example, is probably the most subtle one. In the very last episode that aired last monday, fin travels to mars, and as he's flying out of the atmosphere, for about a half a second you can see a massive chunk of the earth completely gone, as if a nuclear war had occurred
So, yeah. Some complex shit going down.