Last was born in London on 3 December 1894; his father was William Last, director of the Science Museum. He was educated at St Paul's School, London and then Lincoln College, Oxford. Starting late at university because of health problems, he obtained a first-class degree in literae humaniores (classics) in 1918.
Last was appointed as a Fellow of St John's College, Oxford in 1919. His interest in the history of Ancient Rome had been sparked at school by the classical historian T. Rice Holmes, who taught at St Paul's, and continued at Lincoln under William Warde Fowler. His interests also broadened into related spheres such as ancient Oriental history. In 1927, he was appointed as a university lecturer in Roman history, and became Camden Professor of Ancient History in 1936, a post that carried with it a fellowship at Brasenose College. During the Second World War he worked at the British codebreaking centre at Bletchley Park, and in 1948 became the Principal of Brasenose, despite his health and the college's poor financial position.
He retired because his health difficulties in 1956. He died, unmarried, on 25 October 1957.