Jasikevičius was born into a family of sportsmen. His mother Rita, a famous Lithuanian handball player, won a silver medal with the Soviet Union women's national handball team during the 1975 World Women's Handball Championship. Following the success in the world championship, Rita was preparing for the 1976 Summer Olympics in Montreal. According to Rita: "Everything was calculated, planned: Olympic Games, then - the increase of family, one year break and then back to sports again. But everything happened differently" and she had to choose between her son and the Olympic Games. She decided to give birth to a child and give up the Olympic dream. Later on, the Soviet squad went on to win the Olympic gold medals in the Montreal Olympics and the national team's head coach shut the door for Rita Jasikevičienė to return into the national team in the future. However, Šarūnas's mother never regretted her decision. Šarūnas remembers that his mother constantly said: "My son will give me back the Olympics".