Eddington was born in St John's Wood in London to Albert Clark Eddington and Frances Mary (née Roberts). The family were Quakers, and Eddington was brought up by his parents with strict family values. He attended Sibford School, Sibford Ferris, Oxfordshire. In 1952, he married Patricia (née Scott).
Having registered as a conscientious objector, Eddington began his acting career as a teenager with Entertainments National Service Association (ENSA) during the Second World War. He worked for a repertory theatre company in Sheffield and made his first TV appearance in 1956 as a regular cast member of The Adventures of Robin Hood. Initially he played minor characters, but in the fourth season (1959–60), he played Will Scarlet. He also did roles in episodes of The Avengers (1963), The Prisoner (1967) and the final episode of The Champions (1969). He also had a supporting role in Hammer Films' The Devil Rides Out (1968) and appeared as a "straight man" (substituting for regular stooge Henry McGee) in a 1976 episode of The Benny Hill Show.
Although he was an actor all his adult life, it was not until Eddington was in his late forties that he became a household name. This happened thanks to The Good Life, first screened by the BBC in 1975. The sitcom focuses on a suburban couple who decide to give up work and become self-sufficient in their suburban garden. Eddington was cast as Jerry Leadbetter, a neighbour of the main characters, and Penelope Keith played his wife, Margo. Originally intended as bit parts, the Leadbetters quickly became essential foils for the two "stars".
Eddington's fame grew further when he played the title role of Jim Hacker in the comedy series Yes Minister (first screened in 1980) and Yes, Prime Minister (1986–88) – said to have been former Prime Minister Margaret Thatcher's favourite TV programme. He was shortlisted four times for the BAFTA award for Best Light Entertainment Performance for the series, but he lost out to his co-star Nigel Hawthorne on each occasion.
Eddington was made a Commander of the Order of the British Empire (CBE) in the 1987 New Year Honours.
Eddington had been diagnosed with a rare form of cancer, known as mycosis fungoides, when he was 28. The ailment was to cause his death eventually, but in the intervening four decades, Eddington and his immediate family kept his condition private. It only became public knowledge in 1994, when Eddington responded to press speculation about his darkening skin and hair loss.
Eddington died in Southwark, London, on 4 November 1995. He was survived by Patricia, his wife of 43 years, their three sons and one daughter.
