Strange was 6 ft 5 in tall and weighed 220 lbs. His first wife was Flora Hooper of Duncan, Oklahoma. They had two daughters, Wynema and Juanita. Strange was married from 1937 to his death in 1973 to his third wife, Minnie Thompson (1911–2004). The couple had one child, Janine Laraine Strange (born 1939). He had Irish and Cherokee descent through his father. In addition, Strange was an eighth generation great grandson of John Rolfe and Pocahontas through his maternal grandfather.
On September 20, 1973, at age 74, Strange died of lung cancer in Los Angeles, California. Singer Eddie Dean, with whom Strange had collaborated on various songs and opening themes for films, sang at Strange's funeral service as a final tribute. Strange is interred at Forest Lawn - Hollywood Hills Cemetery. In 1975, two years after Strange's death, his Gunsmoke costar Buck Taylor named his third son Cooper Glenn Taylor after Strange.