Is it possible to win in Don't Starve?

There is a win condition now; the previous answers are outdated. There is now a story mode campaign, called "Adventure mode", which can be initiated from a normal game by activating Maxwell's Door, which is a large wooden doorlike structure with a hoop on top that is generated in every (new) Sandbox world. Adventure mode has a series of progressively harder levels, but it does appear to have a definite ending sequence.

The game has no defined winning conditions, at least for the moment. The developers have stated many times that they like the game to remain open-ended, with the player himself defining his own goals.

On their roadmap anyway they have planned a Story Mode that will let you discover the secrets of Don't Starve by progressing through increasingly harder worlds. The portals that have been introduced in the last update (at the time of this answer) are a first step towards that goal.

UPDATE: The game now has a winning condition that can be triggered by playing through Adventure Mode.

The aim is to survive as long as possible.

There is no winning, it's just about surviving. Of course you can set your self challenges like in many open games such as seeing how comfortable you can make your survival etc.

UPDATE: The game can now be won by playing through Adventure Mode.