Born in Dallas, Texas, Clements graduated from Highland Park High School in the Dallas suburb of University Park in 1934. Although Clements was an all-state offensive guard on the Highland Park football team, after his father lost his job due to the Great Depression, Clements worked as an oil driller in South Texas after graduating from high school. In the late 1930s, Clements studied engineering at Southern Methodist University before returning to the oil industry. During World War II, Clements served in the United States Army Corps of Engineers.