There are 4 US Post Offices in Greene County serving a total of 16,932 residents equating to 4,233 residents per post office. It's estimated that approximately 27,025 packages are handled across the 4 county post offices annually. Greene County is a county in central Virginia in the eastern United States. As of the 2010 census, the population was 18,403. Its county seat is Stanardsville. Greene County is part of the Charlottesville, VA Metropolitan Statistical Area."