Lack was born in London and educated at Gresham's School, Holt, Norfolk, and Magdalene College, Cambridge, where he studied Natural Sciences. He was the oldest of four children of Harry Lambert Lack MD FRCS, who later became President of the British Medical Association. The name 'Lack' is derived from 'Lock'. His father grew up in a farming family from Norfolk and became a leading ear, nose and throat surgeon at the London Hospital. Although his father had some interest in birds as a boy it does not appear that he influenced David's interest. His mother Kathleen was the daughter of Lt. Col. McNeil Rind of the Indian army. Kathleen's father was Scottish and on her mother's side was part Irish, Greek and Georgian.
Until the age of fifteen, Lack lived in a large house in Devonshire Place, London. By the age of nine, he had learnt the names of most birds and had written out an alphabetically arranged life-list. In 1928, with an essay on 'My favourite birds' he was the national winner of the senior prize (a silver medal) in the Public School Essay Competition, organized by the Royal Society for the Protection of Birds.