He was born in Shreveport to Alfred and Jessie Jewel Sales Cheesman. His mother lived to the age of 103. Beynon graduated from Byrd High School in 1932, attended Oklahoma City University for one semester, and then enrolled at Centenary College. He quit after his second year to work as an office boy at Arkansas Gas, where later he was promoted to refinery sales. Meanwhile, he took geology courses in night school at Centenary College. Leaving Arkansas Gas he enrolled at LSU in Baton Rouge where he majored in geology, graduated in 1939, and went to work for Atlantic Refining Company in Corpus Christi. He entered the U.S. Army Air Forces as a communications cadet, and trained at Scott Field in Illinois and at Yale University. He was eventually sent to an airbase at Malir, India, near Karachi, serving as communications officer while his unit trained Chinese airmen in flying B-25s. Later Beynon helped build a base in Burma, and served extra duty as mess officer. After the war he returned to Atlantic Refining Company, this time in Midland, Texas, where he took core samples and cuttings of wells being drilled. He was transferred to Jackson, Mississippi, and then to Tulsa. Called up during the Korean War, he spent most of his active duty time at Lockbourne Air Force Base in Columbus, Ohio, where he served as supply officer. Discharged in July of 1952, he returned to Atlantic briefly, then was hired at Norton Oil Company where he remained for 23 years. He married Kathryn Turner, Centenary's dean of women, in May of 1955. They have two children and one grandchild. He still works as an independent geologist.