Stasys Stonkus played his entire career from 1950-58 with Žalgiris Kaunas. With Žalgiris, Stonkus was the Soviet champion in 1951, adding silver in 1952 and bronzes in 1953-56. He also won the 1953 Soviet Cup and was the Lithuanian SSR champion in 1950, 1952-55 and 1958. Stonkus was on the Soviet national team from 1952-57. He won Olympic silver in 1952 and 1956, was the European champion in 1957, and won bronze at the 1955 European Championships.
After retirement from sports, Stonkus, who had graduated from the Lithuanian State Institute of Physical Education (now Lithuanian Sports University) in 1954, worked his whole life with his alma mater. He was a lecturer until 1961, and from 1961-78 was head of the sports games department. In 1978 Stonkus worked briefly as a vice-rector at the university, before becoming the rector of his alma mater from 1978-90. From 1990-92 he was a professor in the sport games department, and from 1992-2009 was again the head of sport games department. From 1992-96 Stonkus served as vice-president of the Lithuanian Olympic Committee and from 1992-98 also worked as an advisor to the Lithuanian president in sports matters. Since 2007, Stonkus was also the chief editor of the Lithuanian Sports Encyclopaedia.