According to a Questions / Answers session where the game developer answered :
Q: Is there an "ending" at which point it's Game Over?
A: Currently there is an evaluation event after 2 years to determine whether you've earned the title of "Stardew Hero". It's not really that important, though... you get to keep playing as long as you want afterward. I would love to keep adding more and more content with time. It's definitely my plan if SV ends up being a success! I am already moving the game towards an "endless" system (e.g. dynamic personalities/schedules for the townspeople), and it will only become more like that in the future. There are so many possibilities!
So the answer is no. There is no limit on the number of days or years in the game. You can play as long as you want.
Source : Stardew Valley Wiki