Carney was born in Delaware County, Ohio. His parents were James and Jane (Ostrander) Carney. James died in 1828, making his wife a widow, and it left four young sons without a father. Thomas stayed at home, farming with his mother, until he turned 19 years old. He studied in Berkshire, Ohio. He lived there with an uncle. He worked in mercantile businesses. He created a successful wholesale business in Leavenworth, Kansas. He was elected to the Kansas Legislature in 1861. He married Rebecca Ann Cannady on November 13, 1861.