Eddyville is a home rule-class city in and the county seat of Lyon County, Kentucky, United States. The population was 2,554 at the 2010 cens…Eddyville is a home rule-class city in and the county seat of Lyon County, Kentucky, United States. The population was 2,554 at the 2010 census, up from 2,350 in 2000. The Kentucky State Penitentiary is located at Eddyville. The town is considered a tourist attraction because of its access to nearby Lake Barkley.