Baker was also a great baseball player. He earned a letter all four years of high school in baseball and led his team to the 1959 Oregon School Activities Association state championship.
Football, however, was Baker's most dominant sport. Baker played quarterback and tailback for the Democrats. In his junior and senior seasons, the Democrats were 23-0 and won two Oregon State Athletic Association state championships. As a senior, he threw for 1,261 yards and ran for 438 yards.
On November 27, 1962, he won the Heisman Trophy for his achievements during that 1962 season. He was the first player to win that award west of Texas, and first from the Pacific Northwest. In addition to winning the Heisman, he also won the Maxwell Award and the W.J. Voit Memorial Trophy in 1962, was a consensus first team All-American, was named as the Sports Illustrated Sportsman of the Year, was a Helms Foundation Award recipient and won 14 player of the year awards, including from AP, UPI and The Sporting News. He also played in the 1963 College All-Star Game, the last time the College All-Stars would ever defeat the reigning NFL champion.
Baker's 99-yard run against Villanova in the Liberty Bowl, the only score in Oregon State's 6-0 victory, remains an NCAA record. Because plays from scrimmage can never start from the goal line, the record can never be broken, only tied. He is now retired from law practice, and still living in Portland, Oregon.