Menadue was born in South Australia on 8 February 1935, the son of a Methodist minister, and was raised in that faith. He later converted to Catholicism. He attended 12 schools, and had lived in 17 houses by the age of 22. He graduated from the University of Adelaide in 1956 with a Bachelor of Economics.
From March 1960 to October 1967 Menadue was private secretary to Gough Whitlam, deputy leader of the Labor Opposition in the federal parliament (Whitlam became leader in February 1967). In 1966 Menadue stood unsuccessfully as Labor candidate for the NSW federal seat of Hume. After leaving Whitlam's employ, he moved into the private sector for seven years as general manager at News Limited, publisher of The Australian.
Menadue was head of the Department of Prime Minister and Cabinet from 1974 to 1976, working under prime ministers Whitlam and Malcolm Fraser. He was closely involved in the events of 11 November 1975, when Whitlam was dismissed by the Governor-General.
He was Australian Ambassador to Japan from 1976 to 1980.
Menadue returned to Australia in 1980 to take up the position of Secretary of the Department of Immigration and Ethnic Affairs. In March 1983, he became Secretary of the Department of the Special Minister of State. He was appointed Secretary of the Department of Trade in December 1983.
Menadue was Chief Executive Officer of Qantas from June 1986 to July 1989. He was a Director of Telstra from December 1994 to October 1996, a Director of NSW State Rail Authority from 1996 to 1999, and Chairman of the Australia Japan Foundation from 1991 to 1998.
Menadue is an adviser to several companies. He chaired the NSW Health Council, which reported to the NSW Minister for Health in March 2000 on changes to health services in NSW. He also chaired the SA Generational Health Review which reported to the South Australian Minister for Human Services in May 2003.
In October 1999, Menadue published his autobiography Things You Learn Along the Way. He was the founding Chair of New Matilda (NewMatilda.com), an independent weekly online newsletter which was launched in August 2004. He is the founder and fellow of public-interest think tank, the Centre for Policy Development.
Menadue was made an Officer of the Order of Australia (AO) in 1985 for public service. In 2003 he was awarded the Centenary Medal 'for service to Australian society through public service leadership'. In 1997, he received the Japanese Imperial Award, The Grand Cordon of the Order of the Sacred Treasure (Kun-itto Zuihō-shō), the highest honour awarded to foreigners who are not head of state or head of government.
Menadue's first wife Cynthia née Trowbridge died of cancer in 1984 aged 49. He has remarried, and has four children and ten grandchildren.