There sure was a nuclear war. The story takes place 16 years after the war, and the future is pretty grim — especially for women, who are valued only for their reproductive capacities. Nuclear and biological warfare have polluted vast areas, and the population suffers a sharp decline in viable births and a rise in birth defects.
Was there supposed to be a nuclear war in The Handmaid's Tale? I couldn't tell.
A secretive group of rightwing fundamentalists murders the U.S. President and members of Congress, and disenfranchises women by taking away their money, jobs, and education. The rightwingers set up the Republic of Gilead — a repressively conservative state bent on annihilating homosexuals, abortionists, and religious sects other than their own. They resettle Jews, old women, infertile women, and nonwhite people in radioactive territories.
Fertile women involved in illicit liaisons or second marriages are apprehended, indoctrinated, and become "Handmaids." These red-uniformed breeders live in seclusion and virtual slavery. They're deprived of their real names and labeled with a patronym of the men who control their lives — as in "Ofcharles" and "Ofwarren." The purpose of these polygamous relationships is the perpetuation of the white race, which carries on warfare in outlying areas in a struggle for supremacy.