Crowe's paternal grandfather, John Doubleday Crowe, was from Wrexham, Wales, while one of Crowe's maternal great-great-grandmothers was Māori. Crowe also has English, German, Irish, Italian, Norwegian, Scottish, Swedish, and Welsh ancestry.
Crowe was born on 7 April 1964, in the Wellington suburb of Strathmore Park, the son of Jocelyn Yvonne (née Wemyss) and John Alexander Crowe, both of whom were film set caterers; his father also managed a hotel. Crowe's maternal grandfather, Stan Wemyss, was a cinematographer who was named an MBE for filming footage of World War II. He is a cousin of former New Zealand cricket captains Martin Crowe and Jeff Crowe, and nephew of cricketer Dave Crowe. Russell has built a cricket field named for his uncle.
When Crowe was four years old, his family moved to Sydney, Australia, where his parents pursued a career in set catering. The producer of the Australian TV series Spyforce was his mother's godfather, and Crowe (at age five or six) was hired for a line of dialogue in one episode, opposite series star Jack Thompson (in 1994 Thompson played the father of Crowe's character in The Sum of Us). Crowe also appeared briefly in the serial The Young Doctors.