Barstow was born in York and was educated at Clifton College and Emmanuel College, Cambridge. He then entered the Civil Service. In 1909 he was appointed a principal clerk in HM Treasury and shortly afterwards became an Assistant Secretary. He was appointed Companion of the Order of the Bath (CB) in 1914.
In 1919 he was appointed Controller of Supply Services at the Treasury, serving in the post until 1927. He was appointed Knight Commander of the Order of the Bath (KCB) in the 1920 New Year Honours. In 1927 he became government director of the Anglo-Persian Oil Company (later the Anglo-Iranian Oil Company), holding the post until 1946.
He also joined the board of the Prudential Assurance Company, becoming deputy chairman and then being elected chairman in 1941. He was chairman until 1953.
- Obituary, The Times, 31 January 1966