What are four clues throughout the story "By the Waters of Babylon" that suggest the setting of the story is New York, sometime in the future after...

Four clues to a nuclear event in New York are that the metal is dangerous, there are many Dead Places within eight days, John calls it newyork, and the "gods" died in everyday positions.

The first hint that the story takes place in a post-apocalyptic version of our world is the reference to metal.  We are told that there is metal in the houses of the gods, and that the metal is dangerous.  John’s people salvage the metal. 

It is forbidden to go to any of the Dead Places except to search for metal and then he who touches the metal must be a priest or the son of a priest. Afterwards, both the man and the metal must be purified. 

Clearly, we have metal all over our own houses.  It also makes sense that the metal might have been dangerous at one time.  If some people really do get sick from touching the metal, it might be the result of some nuclear activity.  The fact that some people can touch it without harm means that perhaps an immunity built up among some.

The hints that it was a nuclear event also come from the fact that there are so many Dead Places.  If it was a small-range, minor incident there would not be as many.

It is eight suns' journey to the east and a man passes by many Dead Places. The Forest People are afraid of them but I am not. Once I made my fire on the edge of a Dead Place at night and, next morning, in the dead house, I found a good knife, little rusted. 

John travels for eight days and passes many places.  This would seem to indicate that whatever occurred, it happened over a large area and killed everyone or almost everyone in its wake.  John’s people are descendants of survivors, either from outside the area or the people who were immune somehow.

Once John gets to the city itself, there are clues abound that this is New York City.  First of all, he refers to it as newyork.  He also seems remnants of buildings and technology that we easily recognize.  He also sees a “god” watching television.  It seems that people died where they were, so that points to a nuclear event.  John also realizes that the gods are not really gods at all, but ancestors of his people who all died from some serious catastrophe.