Josef Ćuk competed at the 1960 Roma Olympics in shooting. He was a silver medalist at the 1954 World Championships with the Yugoslav team in standard rifle, 3 positions. Ćuk won three medals at the 1959 Mediterranean Games, with gold in the small-bore rifle, standing, and the mobile silhouette rifle, adding a bronze in the prone small-bore event. He later became a shooting coach and worked with the Croatian Shooting Federation. In 2010 Ćuk was given the Franjo Bucar National Award for his work and contributions to shooting sport in Croatia.