Eustace was born in Stocksbridge, West Riding of Yorkshire, and began his career as a trainee with Sheffield Wednesday. He made his debut in the 1962–63 season, and played more than 200 games in all competitions, before being sold to West Ham United for a club record £90,000 fee.
He played in midfield, earlier in his career at wing half, later at inside forward, who both made and scored goals. He also played for Rotherham United, Sheffield Wednesday again, and Peterborough United.