Born in Darfield, West Riding of Yorkshire, Storey worked at a W. H. Smith in Barnsley, West Riding of Yorkshire at the age of 14 before becoming a miner at Houghton Main Colliery. Meanwhile, he played non-League football for local colliery sides Ardsley Welfare and Grimethorpe Athletic before signing for Huddersfield Town in September 1943. He was released by the club soon after and returned to play non-League football for Wombwell Athletic, where his father worked as a trainer. He joined Third Division North team York City for a fee of £100 in May 1947, combining mining with playing and training for the club. He was later rated as "perhaps the best bargain York City ever bought". He made his debut in a 2–2 home draw at Bootham Crescent with Wrexham on 27 May 1947 and finished the remainder of the 1946–47 season with three appearances. He missed one game for York during the 1947–48 season, making 42 appearances and scoring 12 goals in all competitions. An injury resulted in him missing the FA Cup semi-final against Newcastle United on 26 March 1955 and several league games, with many fans believing this cost the club promotion, although he was able to play in the 2–0 defeat to Newcastle in the semi-final replay on 30 March at Roker Park.