Baum spent his youth playing for BSG Aktivist Zwenkau and Lokomotive Leipzig. Between 1974 and 1979 Baum played for Sachsen Leipzig and by the 1978–79 season was a regular first-team player. On 28 February 1979 he gained his first cap for East Germany in a 1–0 loss to Bulgaria. The same year saw Baum transfer to city rivals Lokomotiv Leipzig, where he played for the next ten years. Following the transfer, he won the FDGB Pokal three times and played the European Cup. He was part of the team who reached the final of the 1986–87 UEFA Cup Winners' Cup in Athens, losing 1–0 to AFC Ajax.
In total, Baum played in 248 games in the DDR-Oberliga, scoring ten goals. Between 1979 and 1984 he made 17 appearanced for East Germany, winning a silver medal in the 1980 Olympics in Moscow.
After leaving Lokomotiv Leipzig in 1989, Baum had spells at Chemie Böhlen, Sachsen Leipzig and VfB Zwenkau.
Baum became first-team player-coach at SV LVB Leipzig and also coached Kickers 94 Markkleeberg in 1995. Currently, he is coaching the men's first team of the Leipzig Transport Authority team in the Leipzig District League.