Grenoside, Sheffield born goalkeeper Arnold Birch began his football career with Grenoside Bible Class in 1912 and played for Tankersley Colliery in 1913 before the First World War. He then voluntarily joined the 1st Royal Naval Brigade. After being captured during the retreat from Antwerp, Belgium with his division in October 1914, he spent the First World War in a prisoner of war camp in Groningen, Netherlands playing football in the highly rated internal competition alongside Harry Waites. He joined local side Be Quick in 1916 as a masseur and later as coach and was allowed to play for the team in 1918. Be Quick won the Eerste Klasse North and qualified for the Championship play-off finishing fifth.
After his liberation he returned to England and joined First Division Sheffield Wednesday in 1919, making his Football League debut against Middlesbrough that August and he held the gloves until being dropped after 23 appearances following an FA Cup defeat to North Eastern League club Darlington in January 1920, losing his place to Teddy Davison as Wednesday suffered relegation at the end of the season, and he remained Davison’s understudy for three more seasons, making only 6 more appearances before transferring to Third Division (North) club Chesterfield in the 1923 close season.
Birch enjoyed four seasons at Chesterfield and was an ever present in the first two seasons, finishing third in the League in his first season when unusually Birch scored 5 goals, all from the penalty spot, including two on successive days in matches against Durham City and Wigan Borough in April 1924, but after 25 games into the 1926-27 season he again lost his place after an FA Cup defeat to Fulham in January 1927, this time to Bernard Bilcliff, and he left Chesterfield that summer without regaining the gloves having made 152 appearances for The Spireites, joining Midland League club Denaby United. He played two seasons for Denaby before joining Grenoside Sports in 1929 before his eventual retirement.