When Anthony Mancinelli became a barber nearly a century ago, Warren Harding was in the White House, a haircut cost 25 cents and leeches were still used to treat high blood pressure. As hairstyles changed over the decades, he adapted. “I cut them all,’’ he told The New York Times last year, “long hair, short hair, whatever was in style — the shag, the Buster Brown, straight bangs, permanents.” “I have some customers, I cut their father, grandfather and great-grandfather — four generations,” he added. Mr. Mancinelli never stopped cutting hair — except to join the Army and serve in...