Having started his career with Bristol Rovers he moved to Crystal Palace where he became the first £1million goalkeeper in British football and was a member of the Palace side that lost the 1990 Cup Final and won the Full Members Cup in 1991. Subsequently, Martyn spent six seasons at Leeds United. He went on to win 23 England caps. An ankle injury forced him to retire in 2006, following three seasons at Everton.
Martyn made his debut for the England national side against the Commonwealth of Independent States in Moscow in 1992, becoming one of the few Cornishmen to play for England. He earned 23 caps for his national side, spending the peak of his career as second-choice goalkeeper behind David Seaman.
Martyn replaced the injured Seaman for England’s final UEFA Euro 2000 group match against Romania, making an error for Dorinel Munteanu’s equalising goal in an eventual 3–2 defeat. He also started in goal for a 2–2 draw with Greece at Old Trafford that qualified England for the 2002 FIFA World Cup. In Sven-Göran Eriksson’s first game in charge of England against Spain in February 2001, Martyn came on as a substitute for David James and saved a Javi Moreno penalty in a 3-0 win.
Martyn spent time as goalkeeping coach at Bradford City, a role he started in March 2007 as a favour for former Leeds United teammate David Wetherall, who was then caretaker manager at Bradford.