And the answer is "yes," but it wasn't a public school --- the story and a half building there was the final home of Myra Dungan's Chariton Business College, a lively place still during the early 1930s and known by that name as late as the 1940s when it was used to identify the building Ray Coons fell off of while re-roofing.
The photo is of Miss Myra and one of her classes --- the students are not identified --- and was taken I'm guessing not long before World War I --- although I could be mistaken about that.
The college, founded as Chariton Commercial College in 1903-04 by J.W. O'Brian and taken over by Miss Dungan about 1914, operated at various locations. Its first quarters were in the Lincoln Theater/Knights of Pythias building on the south side of the square. When Myra took over, it was across South Main Street from the location of the new post office. The advertisement here was published in The Chariton Leader during the fall of 1914.
Dungan seems to have closed the college during the 1930s (and its building returned to residential use) as she grew older, the Great Depression deepened and programming at Chariton High School and Chariton Junior College became more extensive.
But when the college opened, it filled a need --- and was financed in large part by a business community that needed young men and women equipped with the specific skills necessary to work as clerks, bookkeepers and in other support positions. Myra, one of the original four faculty members, headed up the Normal Department, preparing scholars for the county superintendent of school's examination that, if passed, would qualify them to teach.
Myra, born in 1871, died during 1966 and soon thereafter the last surviving sister, Mary Edna (Dungan) Culbertson --- who had disposed of a troublesome spouse through divorce many years earlier --- closed out the family home and many of its contents came to the newly organized Lucas County Historical Society.
Edna died during 1975, just short of her 100th birthday. With the exception of sister Myrtle, buried in Ottumwa with her husband, Francis Marion Hunter, the Dungan siblings are shoe-horned into the east half of the Dungan-Proctor lot in the Chariton Cemetery, their graves marked only by simple headstones.