There are 4 US Post Offices in Johnson County serving a total of 18,676 residents equating to 4,669 residents per post office. It's estimated that approximately 29,809 packages are handled across the 4 county post offices annually. Johnson County is a county located in the U.S. state of Tennessee. As of the 2010 census, the population was 18,244. Its county seat is Mountain City. It is the state's northeasternmost county, sharing borders with Virginia and North Carolina."