In 1940, Earl S. Nunn and wife Nellie were living in Stanislaus Co., California with children Carmelita (8) and Shelby (4). This suggests a marriage year of about 1931. Per the census, Earl was 26 [therefore b abt 1914], born in Missouri; Nellie was 25 [b 1915], born in Arkansas. [Do you know what her maiden name was?] Earl was a laborer in building construction. This same census indicates that in 1935, they were living in Craigshead, Arkansas.
I find no obvious candidate for him in the 1930 census. Of possible interest is a Joseph E. Nunn (age 53) family in Stanislau Co., California-- might these have been relations and Earl went to the same county by 1940 because he had family there? This entire family was born in Missouri also. Joseph is old enough to be Earl's father (or uncle?)
1920 census is more promising; there is a 7-year-old Earl Nunn, born Missouri, living with parents and siblings in Pike, Stoddard Co., Missouri; parents are Elnore [or Elmer] [m] Nunn (38, born Kentucky) and Lillian (39, born TN). Father was a farmer on his "own place". (i.e., a family farm)
That's all I can do for now. Perhaps someone else can pick up searching for Elmer in 1910 and back.