Unfortunately, no. Staatsburg's population at the last census was about 377 and there are not many domiciles to begin with. As such, I've checked out the rental listings and only one comes up for the whole hamlet. Granted, it's reasonable ($900/month), but your options are going to be extremely limited in Staatsburg proper. There are more in Rhinebeck to the north and the center of Hyde Park to the south.
If you are lucky enough to find one, it'd be rented out of a home. There are no apartment complexes here as far as I know and there definitely aren't apartment buildings with doormen and such like you'd see in New York City.
