The son of Sir William James Bull, MP (1863–1931), who was created a Baronet in 1922, and his wife Lillian Hester Brandon, Anthony Bull was educated at Gresham's School, Holt (1922–1926), and Magdalene College, Cambridge (1926–1929).
The eldest, Sir Stephen John Bull, 2nd Baronet (1904–1942), was killed on active service in Java, East Indies.
After leaving the army, Bull returned to his career with London Transport as its chief Staff and Welfare Officer.
They had one daughter, Caroline (born 11 July 1947), but Barbara Bull died of polio on 18 October 1947.
Anthony Bull joined London's Oxford and Cambridge Club in 1930 and was a member for 74 years.