I would like to know if it's possible to win in Don't Starve? Or is it just the matter of how long until you die?
There is a win condition now; the previous answers are outdated.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.
Building and activating the Teleportato in Survival mode is also a win, of sorts. It is the same device as the one you create in Adventure mode, but the outcome is different.
Putting together the crank thing, the metal potato thing, the box thing, and the ring thing with the wooden thing creates a device that allows you to leave the world you're in.
Upon activation, the game tells you that you'll be destroying the world you're currently playing (and it gives you the opportunity to keep playing instead). And, if you choose to activate it, it gives you the message that "You've escaped!"
At this point, you are brought to the end of game screen where you accrue any owed experience points, and unlock any characters you are now due. And, you are given the opportunity to change to a new character, and begin a new game maintaining your current inventory, but with a new world map and starting Day 1.
To my mind, this does represent a "win" scenario.
Yes, there is an end. If you beat Maxwell in Maxwell's door there is a epilogue (spoiler alert):
If you win you can put the Divining Rod into the Nightmare Lock. Maxwell dies, but ends up in Survival mode alive and Wilson (or the player you are playing) will be trapped in Maxwell's place until another player takes his spot forever.
To me. Winning is surviving survival mode until you get to the portal (or build your own) and proceeded to the next world. Once you've beaten Maxwell and gotten to the select another character screen it's pretty much telling you that you've beaten the main game. However there are secrets that you may have missed that can be unlocked through starting a new game or continuing after you've gone through all the world's and beaten Maxwell.