Plaa was born in San Francisco on May 15, 1930, the son of Jean and Lucienne Plaa, who were Basque French immigrants. He graduated from the University of California in 1952 with a bachelor's degree in criminalistics. The previous year he had married his wife, Colleen Neva; the couple would have eight children. After a brief interlude during which he served in Korea with the U.S. Army Reserve, he returned to complete graduate studies. Unable to obtain a stipend, he was offered to reorient toward pharmacology and toxicology, which he did, earning his M.Sc. in 1956 and his Ph.D. in 1958.