Johnston was born in Woodstock, Georgia, to Florine and Smith Lemon Johnston. His family had been in the area since shortly after the Cherokee were forced out during the Trail of Tears. Johnstons paternal grandfather, who had the given name Doctor Medicine Johnston Jr, owned a general store but believed that education was a waste of time. Johnstons father wanted to become a minister, but he could only afford to attend college briefly before acceding to his family's demands to help them run the store.
Johnston, who was one of four sons, lived on a Georgia farm when he was young. In the early 1930s, Johnston contracted rheumatic fever and the illness affected his heart. A physician uncle told Johnston's father not to send Johnston to college because the young man would not survive long enough to get much use out of the education. Johnston said he later learned that the disease was associated with an average survival period of fifteen years at the time. Florine and Smith Johnston valued education for their children, however, and they sent all of their sons to college.
After going off to Emory University with aspirations of becoming a journalist, Johnston soon realized that the U.S. was headed toward World War II and that a science degree would serve him better. Johnston completed an undergraduate degree in chemistry and a minor in English literature. He received a Ph.D. in chemistry and physics from the California Institute of Technology. As a doctoral student, Johnston focused on the interaction of ozone and the pollutant nitrogen dioxide. While at Caltech, he joined in a secret defense project that involved protecting the country against the use of gas warfare.