Born in Welch, Oklahoma, Bauman debuted in pro ball with Newport in the Northeast Arkansas League. Hitting only three home runs in 59 games with Newport, he also went 0-10 when he was called up to Little Rock in the Southern Association. During the winter, when World War II began, Bauman played semi-pro ball in , and was in the service from 1943 to 1945.
Upon his return, Bauman settled in with Amarillo in the West Texas-New Mexico League. He led the circuit with 48 home runs, 159 rbi and a .301 batting average. In the following season, his home run totals went down, but his production went up. He hit just 38 homers, but he hit .350 and drew 151 walks, and was signed by the Boston Braves.
In , Bauman played in the Braves organization, going 0 for 1 in Class AAA, and posting fair stats in Class AA (.275, 55 BB, 10 HR in 276 AB), while splitting time with Ray Sanders. It was Bauman's only time outside of the high minors, and left it inconclusive as to whether he could play in the majors or not.
Bauman returned to Oklahoma in 1949, signing on for three seasons with the semi-pro Elk City Elks. He also opened a Texaco service station on busy U.S. Highway 66 with a business parter, Jack Riley. The Texaco station was a huge success and the experience served him well as a Texaco dealer in later life.
The Elks did well, especially in 1949 and 1950, and Bauman was a crowd favorite, known simply as "Joe". Fans from western Oklahoma and the Texas Panhandle flocked to watch him knock home runs over the unique red rock wall of Ackley Park. By 1951, the oil-boom economy and the team has begun to sag, and Bauman decided to move on.
At 30, Bauman joined up with the Class C Longhorn League for , signing with Artesia. In that year, his triple crown stats were .375-50-157, good enough to lead the league in homers, RBI, and walks (148). The next year, he led the league in walks (130), runs (135), and home runs (53), while maintaining a high average. After the season, he moved to Roswell.
In , Bauman broke out (if one can "break out" from a 53-home run season), winning the triple crown and leading the league in runs and walks for Roswell. His totals were eye-popping. In 138 games, he had 199 hits in 498 at bats for a .400 average. He hit 35 doubles, 3 triples, and 72 home runs, a record that stood for pro ball until Barry Bonds topped it in 2001. He also drove in 228 runs, and walked 150 times.
Bauman could not duplicate his 1954 season in , hitting only 46 home runs, and batted .336. The following season, he played just 52 games and hit 17 homers. He retired in at age, 34. The career ledger for Joe Bauman reads 1,019 games, 982 runs, 1,166 hits, 337 home runs, 1,057 rbi, 974 walks, and a .337 batting average. He was a true legend on a small stage.
After his baseball career, Bauman continued to run the Texaco station which he had started operating during the last years of his playing career.