Barton was born in South Australia in March 1936; the exact date is unknown. He was born at a siding on the east-west Trans-Australian Railway line, about east of Ooldea. His mother was a Pitjantjatjara woman from the Maralinga lands. It is not known who his father was, and Barton never knew him, but he is believed to have been a White railway worker. He was given the surname "Barton" because that was the name of the siding where he was born (Barton Siding). Archie spent his infancy in the care of his mother at Ooldea, which at the time was a mission station for Aboriginal people.