His first Olympics was in 1928, where he took silver in the team competition. Switzerland won the competition. Individually, he won no medals, but he finished fourth in the horizontal bar, and eighth in the rings. At the next Olympics, in 1932, he again won no medals, but the team finished fourth in the team competition.
At the World Championships, he took three golds in the team competition; in 1926, 1930 and 1938. He also took a team silver in 1934. Individually, he took two silver medals in 1926. In 1930, he took overall silver, behind Josip Primožič, and also won bronze in the pommel horse and rings.
From 15–16 November 2008, an international gymnastics event was arranged in memory of Gajdoš; Jan Gajdoš Memorial 2008 in Brno.