Massey was born in Thakeham, West Sussex, England, the daughter of British actress Adrianne Allen and Canadian-born Hollywood actor Raymond Massey. Her late brother, Daniel Massey, was also an actor. She is the niece of Vincent Massey, a Governor General of Canada.
Massey was on stage as a teenager and made her film debut in 1958. In 1960, she played a potential murder victim in Michael Powell's cult thriller Peeping Tom. She played the role of the cockney barmaid Babs in Alfred Hitchcock's Frenzy (1972). She appeared with her brother Daniel playing deadly siblings in the 1973 horror film The Vault of Horror.
Massey continued to make occasional film and stage appearances, but has worked more frequently in television, in dramas such as The Pallisers (1974), the 1978 adaptation of Rebecca (in which she starred along with her ex-husband, Jeremy Brett), The Mayor of Casterbridge (1978), The Cherry Orchard (1980), and Anna Karenina (1985). She had roles in the British comedy series The Darling Buds of May (1991) and The Robinsons (2005). She has also appeared in a number of mysteries and thrillers on British television, including episodes of Inspector Morse, The Inspector Alleyn Mysteries, Midsomer Murders, Strange, and Lewis. She is the narrator of This Sceptred Isle on BBC Radio. She also played the part of Josephine Daunt in Daunt and Dervish on BBC radio.
In 1986, Massey was awarded the British Academy TV Award for Best Actress for her role in Hotel du Lac.She also appears as Mrs. D'Urberville in the 2008 BBC adaptation of Tess of the D'Urbervilles, an older version of May and as Rosie in An Angel For May, and in the 2004 BBC version of Our Mutual Friend.
In the New Year's Honours List published 31st December 2004 she was created a Commander of the Order of the British Empire for services to drama.
She published an autobiography in 2006, Telling Some Tales revealing a difficult early life, with her father Raymond Massey fleeing the family for the United States when Anna was little, and re-marrying, while her mother, Adrianne, is remembered as a marvellous hostess who filled the house with an exotic mixture of guests. But it was her Nanny who supplied the crucial emotional support, and who remained with Anna until her death. Massey writes of her psychotherapy which she says helped get her through her troubles, and of the peaks and valleys of her failed marriage to actor Jeremy Brett (who struggled with bipolar disorder), and of their son, writer David Raymond William Huggins (b.1959), and of her successful second marriage to Uri Andres, and of being a grandmother to Dan (b.2002). In spite of the vicissitudes of Massey's life, Telling Some Tales becomes a story of personal happiness, achieved 'without makeup, lighting or script'.
Of life today, Massey has been quoted as saying, "Theatre eats up too much of your family life. I have a grandson and a husband and I'd rather I was able to be a granny and a wife."