Burkitt was born in Enniskillen, County Fermanagh, Ireland. He was the son of James Parsons Burkitt, a civil engineer. Aged eleven he lost his right eye in an accident. He attended Portora Royal School in Enniskillen and Dean Close School, England. In 1929 Burkitt entered Trinity College, Dublin, to study engineering, but believing his evangelical calling was to be a doctor, he transferred to medicine. In 1938 he passed the Royal College of Surgeons of Edinburgh fellowship examinations. On 28 July 1943 he married Olive Rogers.