Ounapuu was born in Sweden to an Estonian family, and moved to Canada in 1950. He was an Assistant Crown Attorney in York, Ontario during the 1970s, and was appointed by provincial Attorney General Roy McMurtry to shut down the unlicensed body-rub parlours on Yonge Street in 1977. He was successful, saying that "there wasn't a bawdy house left" when he finished his work. Ounapuu took a two-year leave of absence to work as a prosecutor in Hamilton, Bermuda in 1978, and became a criminal lawyer following his return to Canada. He has participated in several notable cases, and as of 2006 is representing former Toronto City Councillor Steve Ellis against high-profiles charges of misconduct.