Jacobi, an only child, was born in Leytonstone, London, England, the son of Daisy Gertrude (née Masters), a secretary who worked in a drapery store in Leyton High Street, and Alfred George Jacobi, who ran a sweet shop and was a tobacconist in Chingford.

His great-grandfather emigrated to England from Germany during the 19th century. His family was working class. Although a war baby, he claims a happy childhood. In his teens he went to the Leyton County High School and became an integral part of the drama club, The Players of Leyton.

Jacobi quickly came to the fore, and his talent was recognised by Laurence Olivier, who invited him back home to London to become one of the eight founding members of the new National Theatre, even though at the time he was relatively unknown.

He played Laertes in the National Theatre's inaugural production of Hamlet opposite Peter O'Toole in 1963, and Olivier gave him the role of Cassio in his 1965 film of Othello and of Andrei in Three Sisters in 1970.

After eight years at the National Theatre, Jacobi left in 1971 to pursue different roles and mediums of expression. In 1972, he starred in the BBC serial Man of Straw, directed by Herbert Wise.

Most of his theatrical work in the 70's was with the touring classical Prospect Theatre Company, with which he undertook many roles, including Ivanov, Pericles, Prince of Tyre and A Month in the Country opposite Dorothy Tutin (1976).

Although Jacobi's name was becoming known and he was increasingly busy with stage and screen acting, his big breakthrough did not come until 1976. It was the title role of the BBC's blockbuster series I, Claudius that finally cemented his increasing reputation with his performance as the stammering, twitching Emperor Claudius winning him many plaudits.

In 1979, thanks to his international popularity he took Hamlet on an epic theatrical world tour through England, Egypt, Greece, Sweden, Australia, Japan and China with himself in the protagonist's role. He was then invited to essay the role once more at Kronborg Castle, better known as Elsinore Castle, the setting of the play itself. In 1978 he played in the BBC's production of Shakespeare's Richard II, with Sir John Gielgud and Dame Wendy Hiller.