There are 5 US Post Offices in Gregory County serving a total of 4,086 residents equating to 817 residents per post office. It's estimated that approximately 6,522 packages are handled across the 5 county post offices annually. Gregory County is a county located in the U.S. state of South Dakota. As of the 2010 census, the population was 4,271. Its county seat is Burke. The county was created in 1862 and organized in 1898. It was named for the politician J. Shaw Gregory."