He attended an Anglican boarding school near Perth, and in February 1944 enslisted in the Royal Australian Navy during World War II.
At the time of his discharge on 29 May 1956, he held the rank of Lieutenant in the RANVR Special Branch and was stationed at the shore training base HMAS Cerberus.
He was head of the Australian stations in Singapore, Cambodia and South Vietnam, and later rose to the position of Deputy Director-General of ASIS.
On 20 April 1983, Barnett requested and was granted an urgent meeting with the newly elected Prime Minister, Bob Hawke, whereupon he briefed him on what he called a 'matter of national security'—ASIO was aware of several meetings between David Combe, the national secretary of the Australian Labor Party, and First Secretary of the Soviet Embassy in Australia, Valery Ivanov.
Ivanov was expelled from Australia, and although the second Hope Royal Commission in 1984 established that attempts were being made to recruit Combe, no intelligence breaches had taken place.