Barnett, from a mining family, worked down Hanley Deep Pit at the age of fifteen. Expected to start a mining career, he instead impressed playing football at Canon Street School and Stoke Boys' Club, and won an amateur contract at Port Vale in 1954. During his national service he played alongside Ken Higgs in the Army Medical Corps football side. He signed professionally for Port Vale in June 1956, and won a place in Freddie Steele's first team after scoring eighteen goals in seventeen games for the reserves and also scoring twice past Middlesbrough's Rolando Ugolini in a friendly for the first team. He scored on his debut at inside-left in a 4–2 defeat to Millwall at The Den on 13 December 1958. Manager Norman Low described him as the 'supreme goal poacher' as he netted 20 goals in 22 appearances in what was left of the season, helping the club to win the Fourth Division title. Despite this he did not get on with Low, and had many rows with his manager. He was the club's top scorer during the 1959–60 season with 17 goals in 35 games, including four against Halifax Town in a 7–0 win at Vale Park on 28 December. He also grew a beard and refused to shave until Vale were knocked out of the FA Cup, doing so after defeat to Aston Villa at the fifth round stage.
Tranmere Rovers
He was sold to Peter Farrell's Tranmere Rovers for a £5,000 fee in March 1960. Tranmere were heading for relegation into the Fourth Division, but Barnett helped them to maintain their Third Division status by the end of the season. The "Superwhites" then failed to avoid relegation in the 1960–61 season under the stewardship of Walter Galbraith, and Barnett moved away from Prenton Park to Halifax Town in August 1961.
Halifax
He scored nine goals in 32 Third Division appearances for Halifax in the 1961–62 campaign.
Barnett returned to England and joined Cheshire County League side Macclesfield Town, scoring two goals on his debut in a 3–3 draw with Witton Albion at Moss Rose on 1 January 1966. He then scored another brace against Wrexham in his next appearance, and would score a total of six goals in ten games for the "Silkmen".
Later life
He returned to England in 1971 and coached Vale's "A" team, later working in the club's commercial department and running the Y.T.S. team until 1985, at which point he became a newsagent. He and wife Ena and went on to have three children, seven grandchildren, and seven great-grandchildren.