Leavelle was born and raised in Red River County, Texas. In 1937, Leavelle joined the Civilian Conservation Corps. Following graduation from high school, Leavelle joined the United States Navy in 1939 (during World War II) and served as a sailor on board the USS Whitney; he was on board the ship during the Japanese attack on Pearl Harbor on December 7, 1941. In April 1942, the Whitney left Pearl Harbor and headed for the South Pacific Ocean to support operations there. After being knocked down and seriously injured by a rogue wave, he was evacuated to Oak Knoll Naval Hospital near Oakland, California. In the hospital, he met nurse Taimi Snelma, his wife-to-be. Around the same time, Leavelle left on a medical discharge to take a civilian supply job with the Army Air Force in Southern California.