Bican was born in Vienna to František and Ludmila Bican. He was the second of three children. František was from Sedlice in Southern Bohemia, and Ludmila was Viennese Czech. Josef’s father František was a footballer who played for Hertha Vienna. He fought in World War I and returned uninjured. However, František was to die at the age of just 30 during 1921 because he refused an operation to treat a kidney injury sustained in a football match. His mother worked in a restaurant kitchen. The family’s poverty meant that Bican had to play football without shoes, which helped him improve his ball control skills. Bican attended the Jan Amos Komenský school, a Czech school in Vienna. Four years after his father’s death in 1925, twelve-year-old Bican started to play for the Hertha Vienna junior team, Hertha Vienna II. When he was 18, Bican was spotted by Rapid Vienna, who were a big club in the city at the time.

On 29 November 1933, aged 20 years and 64 days, Bican made his debut for Austria in a 2–2 draw against Scotland. He later played for them at the 1934 World Cup, when the Austrian Wunderteam reached the semifinals. His solitary goal in the tournament came in extra-time in the Austrians’ 3–2 win over France.

During 1937, Bican left Vienna to join Czech club Slavia Prague. He played for Slavia throughout World War II, while many of his football rivals were at war. During eight league seasons he would score 328 goals, including 57 in 26 matches one particular season. Three times in his career, Bican scored seven goals in a game. In a 1939–40 league match against Zlín, Bican found the net seven times as Slavia ran out 10–1 winners. During the 1940–41 season, Bican matched his feat of the previous season, again against Zlín, scoring seven times, as Slavia won by a 12–1 scoreline. It was 1947–48 before Bican managed his third seven-goal match, as Slavia defeated České Budějovice in a game which finished 15–1.

At the time Bican was playing for Slavia Prague, he applied for Czechoslovak citizenship. However, when he eventually became a Czechoslovak citizen, he discovered that a clerical error meant he couldn’t play at the World Cup during 1938. In total, he scored 34 goals in 45 international matches for 3 teams (Austria, Czechoslovakia and Bohemia & Moravia). His final national team appearance was for Czechoslovakia in a 3–1 defeat against Bulgaria on 4 September 1949.

Bican tried to improve his standing with the Communists by joining steel works Železárny Vítkovice. During 1951, he joined FC Hradec Králové, but, on 1 May 1953, the Communist Party forced him to leave the city and, therefore, the club. After being forced to leave, he returned to Slavia Prague, or, as it was known then, Dynamo Prague. He finally retired from playing, still at Slavia, at the age of 42 during 1955. He was the oldest player in the league at that time.

Around this time, Pelé was getting ready for his 1000th goal and many journalists were searching for another player who had scored a thousand goals. Former Austrian player Franz “Bimbo” Binder suggested Bican, who he claimed had scored 5000 goals. When reporters asked Bican why he hadn’t made more of a fuss about his goalscoring feats, he simply said, “who’d have believed me if I said I’d scored five times as many goals as Pelé?” However, to score this 5000 goals he should have kept an average of 185 goals/year along all his 27 years of career, but only counting goals in official matches, Bican scored at least 805. Bican’s goal-scoring feats are often forgotten because he did not make a big fuss about it in the media. His record is often overshadowed by Pele’s 1283 goal record. But shortly before his death in 2001, IFFHS based on RSSSF statistics declared Bican with 643 league goals, the most prolific scorer of the 20th Century. This was judged by the number of times a player had been top scorer in his domestic league. Bican managed this feat 12 times, more than any other player in football history.