Isaacs was born to Jewish parents in Liverpool on 6 June 1963. His father was a jeweller. He has two older brothers and one younger brother. He spent his earliest childhood years in the Liverpool suburb of Childwall, in a "closely knit & integrated" Jewish community co-founded by his Eastern European Jewish great-grandparents. He has said that being Jewish played a big role in his childhood, as he attended youth club in the local synagogue of King David High School in Liverpool's Childwall district, as well as a cheder twice a week as a young adult. When he was 11, he moved with his family to London and attended the Haberdashers' Aske's Boys' School in nearby Elstree at the same time as David Baddiel, Sacha Baron Cohen, Mark Kermode, and Matt Lucas. He describes the bullying and intolerance he observed and experienced during his childhood as "preparation" for portraying the "unattractive" villains he has most often played.
As a Jewish teen in London, Isaacs endured antisemitism by the National Front, a far-right extremist organisation. His parents eventually emigrated to Israel. He later told an interviewer, "There were constantly people beating us up or smashing windows. If you were ever, say, on a Jewish holiday, identifiably Jewish, there was lots of violence around. But particularly when I was 16, in 1979, the National Front were really taking hold, there were leaflets at school, and Sieg Heiling and people goose-stepping down the road and coming after us." Following in the footsteps of his three brothers (one who became a doctor, one a lawyer, and one an accountant), he studied law at Bristol University from 1982 to 1985, becoming involved in the university's theatre club there; he eventually acted in over 30 plays and performed each summer at the Edinburgh Festival Fringe, first with Bristol University and then twice with the National Student Theatre Company. After graduating, he went immediately to train at London's Central School of Speech and Drama from 1985 to 1988.
