There are 8 US Post Offices in Ohio County serving a total of 19,250 residents equating to 2,406 residents per post office. It's estimated that approximately 30,725 packages are handled across the 8 county post offices annually. Ohio County is a county located in the U.S. state of Kentucky. As of the 2010 census, the population was 23,842. Its county seat is Hartford. The county is named after the Ohio River, which originally formed its northern boundary. It is a dry county, which means that the sale of alcohol is restricted or prohibited."