Barnett played as a goalkeeper and started out at Everton in 1962, winning the 1965 FA Youth Cup and gaining schoolboy and U21 honours with England.

1 goalkeeper Bob Wilson broke his arm early in the season, and Barnett made his debut against Coventry City on 4 October 1969.

Barnett made 11 league appearances and successfully kept a clean sheet over both legs against Sporting CP in the Inter-Cities Fairs Cup Second Round (saving a penalty in the goalless first leg), but once Wilson recovered from his injury, Barnett was relegated to the reserves and Wilson was in goal for the final.

Nevertheless, he stayed loyal to Arsenal and played for another two years as Rimmer's understudy, his last appearance coming against Stoke City on 13 December 1975 (his only match in the 1975/6 season).

The team folded after the 1981 NASL season[2] However, he never pursued a career in management and eventually returned to England.