Storer made his debut against Lincoln City on 1st September 1894. He missed only two games that season. In April 1895 he was chosen to play for the Football League against the Scottish League and therefore was the first Arsenal player to win representative honours.
In 1895 Storer, a right-handed batsman, played five games for Derbyshire County Cricket Club. Storer played in the first twelve league games in the 1895-96 season before being suspended by the club for four weeks in November 1895 for a breach of discipline. He never played for Woolwich Arsenal again as he was sold to Liverpool in December 1895.
Storer made his debut for Liverpool against Manchester City on 1st January 1896. He kept his place for the remaining 11 games and helped the club win promotion to the First Division. He was also the first-choice goalkeeper for the 1897-98 season.
Harry Storer died at Holloway, Derbyshire on 25 April 1908, at the age of 37, from tuberculosis.