Billig was born at 41 Hanbury Street, Spitalfields, in the East End of London, and grew up around Brick Lane where Jewish friends had taken them in. Her parents, Barnett and Milly Billig, were refugees from Russia, escaping from anti-Jewish pogroms. The Billig family had six children, four of which became doctors. Her sister was a nurse and a brother Levi was an oriental studies scholar who was shot dead in Jerusalem in 1936. Parents Barnett and Milly encouraged their children to spend their time reading instead of playing in the streets. Her mother stayed home with the children, managing the household, while Barnett worked as a newsagent rolling cigarettes and eventually as a cigar roller. At the age of 11, Billig received a scholarship to attend Myrdle Street Central School in Stepney. She remained at the top of her class which eventually earned her a scholarship to London University.