Sher was born into a Lithuanian Jewish family in Cape Town, South Africa, the son of Margery and Emmanuel Sher, who worked in business. He grew up in the suburb of Sea Point (his cousin is Ronald Harwood), but he has worked mainly in the United Kingdom and is now a British citizen.

In 1968, after completing his compulsory military service, he left for London to audition at the Central School of Speech and Drama, but was unsuccessful. Instead, he studied at the Webber Douglas Academy of Dramatic Art from 1969 to 1971. After training, and some early performances with the theatre group Gay Sweatshop, he joined the Royal Shakespeare Company in 1982.

In the 70's Sher was part of an astonishing group of young actors and writers working at the Liverpool Everyman. It consisted of the likes of writers Willy Russell and Alan Bleasdale and fellow actors Bernard Hill, Julie Walters, Trevor Eve and Jonathan Pryce.

The work put on was highly regarded (Two successes being "John, Paul, George, Ringo … & Bert" with Sher playing Ringo and "Richard III" with Sher playing Buckingham). Sher summed up the work of the company with the phrase "Anarchy ruled."

In the 80's he worked at The Royal Shakespeare company, taking the title role in "Tartuffe" and playing The Fool in "King Lear" before his big breakthrough in 1984, when he played the title role in Shakespeare's Richard III. This won him the prestigious Laurence Olivier Award. Since then he has played the lead in many big productions, including Tamburlaine, Cyrano de Bergerac, Stanley, Macbeth and The Merchant of Venice.

In 1997, his portrayal of Disraeli in the film Mrs. Brown was well received, and he won his second Laurence Olivier Award for his performance as Stanley Spencer in Stanley.

In television, he starred in the miniseries The History Man (1981) and The Jury (2002). In 2003 he played the central character in an adaptation of the J G Ballard short story The Enormous Space, filmed as Home and broadcast on BBC Four.