Born in Akola, British India in 1908, Barr was educated at Radley College and Trinity College, Oxford, where he rowed in the 1929 Boat Race and was awarded a Blue. He went from stage to screen with The Merry Men of Sherwood (1932). He spent the 1930s playing various beneficent authority figures and "reliable friend" types. As a conscientious objector during the Second World War, Barr helped people in the Blitz in London's East End before serving with the Friends Ambulance Unit in Africa. There he met his wife Anne "Jean" Williams, marrying her after ten days; it would have been sooner, but they needed permission from London.
In 1946, he picked up where he had left off, and in the early 1950s he began working in British television, attaining a popularity greater than he had while playing supporting parts in such films as The Case of the Frightened Lady (1940) and The Blue Lagoon (1949).
This popularity enabled Barr to obtain better roles and command a higher salary for his films of the 1950s and 1960s. Some of the films in which he appeared during this period were The Dam Busters (1955), Room in the House (1955), Saint Joan (1957), Next to Next Time (1960), Billy Liar (1963), The First Great Train Robbery (1979) and Octopussy (1983). On television, he appeared in Doctor Who in 1967 as Hobson in the serial entitled The Moonbase; in the 1970 Randall & Hopkirk (Deceased) episode "You Can Always Find a Fall Guy" and appeared once in The Avengers. In the 1981 BBC Radio 4 adaptation of The Lord of the Rings, Barr voiced the role of Gamling.