"By the Waters of Babylon" definitely registers as post-apocalyptic fiction, and the author Stephen Benet gives clues throughout the story to suggest that John and his family live in a time long after the U.S. has had a nuclear holocaust.  The opening paragraph in the story that references the "Dead Places" and the idea that the people were forbidden to go east makes the reader think that perhaps something terrible has happened in the east not only to wipe out civilization, but make it dangerous to go there.  The logical conclusion is that the east has been destroyed by nuclear war; the post-apocalyptic societies would have forbidden people to go there because of the danger of radiation.  They might not have identified it as such, but they would have quickly realized that the people who traveled into those dead zones were becoming ill; the tribal elders would have to make travel there forbidden in order to protect people. 

Later in the story, the pieces and clues left by Benet quickly begin to come together, especially as the reader sees the Place of the Gods through John's unfamiliar eyes.  He mentions god roads that are "cracked and broken" as well as god towers along with the "marks and stains of the Great Burning."  Although John does not have the vocabulary to describe what he is seeing, the reader can understand and decipher that John views what remains of a fallen city after a nuclear bombing.  He tells the tale of the city's destruction as a terrible fight, for "when gods war with gods, they use weapons we do not know," leaving "poison in the ground."  The details and imagery provided by the author undeniably point to the city's destruction through nuclear weapons.