Carr was born in Thornton in Craven, a village in the West Riding of Yorkshire, England, where his father James was a solicitor. He was educated at Marlborough College in Wiltshire and at Trinity College, Cambridge, where he gained an arts degree in 1904, specialising in classics: he read both Latin and Greek. He then studied medicine at the Royal London Hospital, becoming an LRCP and MRCS in 1908. He remained at London Hospital as a resident medical officer until 1910, and then worked as a ship's doctor on a merchant vessel.