Harris was born on 1 October 1930, at Overdale, 8 Landsdown Villas, Ennis Road, Limerick, and was the fifth in a family of eight children, (six boys and two girls), to flour merchant Ivan Harris and Mildred (née Harty). Overdale was "a tall, elegant, early 19th-century redbrick" house with nine bedrooms, in a wealthy part of Limerick, the houses "built at the turn of the 20th century for Limerick's burgeoning middle class... people who could afford properly grand drawing rooms, a bedroom each for the children and one for the pot, plus space for a few servants". He was educated by the Jesuits at Crescent College. A talented rugby player, he appeared on several Munster Junior and Senior Cup teams for Crescent, and played for Garryowen. Harris's athletic career was cut short when he caught tuberculosis in his teens. He remained an ardent fan of the Munster Rugby and Young Munster teams until his death, attending many of their matches, and there are numerous stories of japes at rugby matches with actors and fellow rugby fans Peter O'Toole and Richard Burton.
After recovering from tuberculosis, Harris moved to England, wanting to become a director. He could not find any suitable training courses, and enrolled in the London Academy of Music and Dramatic Art to learn acting. He had failed an audition at the Royal Academy of Dramatic Art and had been rejected by the Central School of Speech and Drama, because they felt he was too old at 24. While still a student, he rented the tiny "off-West End" Irving Theatre, and there directed his production of Clifford Odets's play Winter Journey (The Country Girl).
After completing his studies at the academy, he joined Joan Littlewood's Theatre Workshop. He began getting roles in West End theatre productions, starting with The Quare Fellow in 1956, a transfer from the Theatre Workshop. He spent nearly a decade in obscurity, learning his profession on stages throughout the UK.