Dunne's father inherited the Woodside Restaurant Temperance Refreshment Rooms and an auctioneering firm, and his mother ran a drapery business and later a shipping agency.
While still in school he took on a number of jobs, including repairing bicycles, rearing sheep, and working as a boot-boy at the home of Sir John Foster George Ross of Bladensburg.
He was employed as an apprentice first in Anderson's of West Street, Drogheda and later in Cameron's Drapery store in Longford.
In the mid 1930s, he moved to Cork, where he worked as a buyer for Roches Stores in the menswear department.
They were married in September 1939 and had six children, including Margaret, Frank, Elizabeth, Therese, and Ben.