There are 31 US Post Offices in Greene County serving a total of 46,919 residents equating to 1,514 residents per post office. It's estimated that approximately 74,888 packages are handled across the 31 county post offices annually. Greene County is a county located in the U.S. state of New York. As of the 2010 census, the population was 49,221. Its county seat is Catskill. The county's name is in honor of the American Revolutionary War general Nathanael Greene."