There are 5 US Post Offices in Martin County serving a total of 10,552 residents equating to 2,110 residents per post office. It's estimated that approximately 16,842 packages are handled across the 5 county post offices annually. Martin County is a county located in the U.S. state of Kentucky. As of the 2010 census, the population was 12,929. Its county seat is Inez. The county was founded in 1870 and is named for Congressman John Preston Martin. It is a prohibition or dry county, which means that alcohol sales are prohibited in all areas of the county.