At any airport which has an airside transit area that is kept open around the clock, a story like in "The Terminal" could theoretically take place.

I'm thinking it would not be possible at most US airports, because the facilities available between the arrival gate and the immigration desks tend to be very basic - there's usually just some restrooms and water fountains, no shops, no restaurants or the like. As such, you couldn't keep yourself going for very long, and that's not even taking into account that the international terminals at most US airports shut down at some point during the night, or the fact that because of America's hypersensitivity towards perceived security concerns, you would be dragged off in shackles once they figure out what you're doing.

In most of the major international airline hubs of the world, I would imagine that, assuming you had enough money to keep yourself fed, you could spend weeks or even months airside at an airport without anyone really noticing or caring, though the question is whether that would be a desirable state of being.