There are 2 US Post Offices in Ouray County serving a total of 3,806 residents equating to 1,903 residents per post office. It's estimated that approximately 6,075 packages are handled across the 2 county post offices annually. Ouray County /\u02c8j\u028a\u0259re\u026a/ is one of the 64 counties of the U.S. state of Colorado. As of the 2010 census, the population was 4,436. The county seat is Ouray. Because of its rugged mountain topography, Ouray County is also known as the Switzerland of America."