Dobinson was born in 1898 in Darlington, County Durham, the son of Thomas William Dobinson, a porter, and his wife Annie. He attended Darlington Grammar School, where he played for the school football team, before joining the Army in 1915. He served in France with 52 Brigade Royal Field Artillery as well as playing for the Brigade's football team. After the war, he worked as a shipyard clerk, played football as an amateur for Sunderland West End, and established a reputation in Yorkshire and the north-east of England as a middle-distance runner.
