Lemons was born in Walters, Oklahoma and given the initials-only name "A.E.". He grew up in the town of Walters, Oklahoma and graduated from Walters High School in Spring 1941.
He earned a basketball scholarship to play for Southwestern Oklahoma Teachers College (now known as Southwestern Oklahoma State University). and their long-time coach Rankin Williams.
After United States entered World War II in December 1941, he joined the Merchant Marine. He served in the Pacific and often referred to the pressures of his war experience to put sports pressures into perspective.
After the war, he enrolled at Hardin College, which had just added a four-year senior college in 1946. He was a 6-foot 4 inch center/forward for the Indians, who finished 4-15 under first-year head coach Fermon “Red” Rutledge, during the 1946-47 season.
In 1947, he transferred to Oklahoma City University (OCU) where he played two years for coach Doyle Parrack. In 1947-48, the Chiefs had an 18-13 record, which improved to 20-6 in 1948-49.
He married Betty Jo Bills, and they had two daughters Dana and Jan.
Lemons spent 25 years as head coach at OCU. His first 18 years at OCU from 1955–1973, during which he led OCU to a 309-181 record, two NIT berths and seven NCAA tournament appearances in 1956, 1957, 1963–1966 and 1973. OCU also won the All-College Tournament in 1965. Lemons coached several All-America & future NBA players, such as Arnold Short and Hub Reed.
From 1973 to 1976 Lemons was head coach at Pan American University, where he was named 1974–75 Texas Coach of the Year and coached the nation's leading scorer in Marshall Rogers.
He took the head coaching position at the University of Texas in 1976. He served as president of the National Association of Basketball Coaches in 1977 and was named National Coach of the Year in 1978. Lemons' last season at Texas was 1981–82. In March 1978, Lemons led the University of Texas to the championship of the NIT with a victory over North Carolina State.
In his second stint at OCU, he took the Chiefs to the NAIA Championship tournament once and to the District IX playoffs four times. Lemons was Sooner Athletic Conference Coach of the Year in 1985-1986. OCU had an undefeated season record and a trip to the NAIA tournament in 1986-1987. That year, they were ranked number one throughout the season. The season ended with a 34-1 record, Lemons was named District 9 Coach of the Year and Sooner Athletic Conference Coach of the Year. In 1987, he was named Basketball Times Coach of the Year. In 1989, he received the Jim Thorpe Lifetime Achievement Award.
During his 25 years with OCU, Lemons posted a record of 432-264. He brought positive national attention to the state of Oklahoma, Oklahoma City and OCU. He established himself as a "teacher of men," not only in sports, but in the values of life, as proven by the success and leadership accomplishments of his students and players. In 1990, Abe was inducted into the Oklahoma Sports Hall of Fame.
Lemons died on September 2, 2002 of complications from Parkinson's Disease at the age of 79.