Brabec was born in Litoměřice, then in Czechoslovakia, and later moved to Prague, where he joined the Dukla Prague athletics club and began competing in shot put competitions. He grew to a height of. He first emerged at national level at the age of twenty winning both the indoor and outdoor national titles. His throw of at that year's Czechoslovak Athletics Championships was a meet record. He threw beyond twenty metres for the event that season and his new best of ranked him tenth globally among shot putters for the year. This made him the first Czechoslovak shot putter to reach the top ten since Jiří Skobla in 1961. At the 1971 European Athletics Championships he did not perform as well, however, failing to make it through the qualifying round. He won his first international medal at the 1972 European Athletics Indoor Championships: following on from Skobla's bronze medal in 1966, Brabec himself took the bronze medal behind Hartmut Briesenick of East Germany and Poland's Władysław Komar. Nationally, Miroslav Janoušek took the indoor title, but Brabec improved his meet record at the outdoor championships with a throw of. He reached his first Olympic final later that year, coming tenth overall. A new national rival – Jaromír Vlk – finished one place ahead of him. At the 1973 European Athletics Indoor Championships gave his best international performance yet with a throw of which brought him the gold medal ahead of East Germany's Gerd Lochmann and fellow Czech Vlk. Brabec became only the third Czechoslovak man to win at that competition, following on from 800 m runner Jozef Plachý and triple jumperPetr Nemšovský. This season proved to be the peak of his career as he achieved a lifetime best mark of at the Czechoslovak Championships – this stood as a record until Remigius Machura bettered it in 1982 with his national title win. He also won a second national title that year. He had his highest ever seasonal ranking in 1973, in fourth place behind Americans Al Feuerbach and George Woods and Germany's Briesenick. In the 1974 season he was usurped at the national championships by Vlk, but managed to win the third medal of his career at the 1974 European Athletics Indoor Championships. He also competed at the 1974 European Athletics Championships, taking seventh place in what was his highest ever finish at that tournament. A national title double in 1976 preceded his second Olympic appearance at the 1976 Montreal Games, at which he ranked eleventh in the final. He had his last three major international outings at the end of the 1970s, taking sixth at the 1978 and 1979 European Athletics Indoor Championships, as well as eighth at the 1978 European Athletics Championships. Brabec continued to be a presence at national level, taking national indoor titles in 1978 and 1979, before having three further outdoor wins in 1979, 1981 and 1982. The rise of World Championships medallist Remigius Machura coincided with the end of Brabec's career at a high level. After retiring from competition, he served as coach for the athletics section of Dukla Prague. Among those he trained were Machura, František Vrbka, Robert Změlík, Vladimíra Racková, Pavel Sedláček, Vladimír Maška and Jan Železný. Brabec died on 20 May 2018.
