Levy was born to a Jewish family in downtown Hamilton, Ontario. His mother, Rebecca (née Kudlatz), was a homemaker, and his father, Joseph, was a foreman at an automobile plant. He has a brother, Fred, and sister, Barbara. His mother was born in Glasgow, Scotland, to Polish Jewish parents, later moving to Canada.

The Canadian actor married screenwriter and producer Deborah Divine in 1977. The couple went on to have two children, Dan and Sarah, whom they raised in Toronto away from a " show-bizzy kind of environment," Eugene told PEOPLE in 2019. And, according to Eugene, he was a strict parent to his son and daughter.