Elkabbach was born to an Algerian Jewish family in Oran in 1937, then the prefecture of the département d'Oran in French Algeria. He began his journalistic career in 1960 as a radio correspondent in Algiers, but having taken part in the strikes of May 1968 he was sidelined and sent to Toulouse. Elkabbach would later spend time in Bonn, Germany before venturing into television news in 1970. From 1993 to 1996 he served as president of France 2 and France 3, from 1999-2009 he was president of the television station Public Sénat, and was at the helm of Europe 1 from 2005-2008.

Elkabbach presents Bibliothèque Médicis on Public Sénat, during which he interviews an eclectic mix of international literati, political leaders, intellectuals and historians.

He is the father of successful actress Emmanuelle Bach.