The specific locations mentioned include:
* The Hatcher's apartment building on the Upper West Side - This is where most of the story's events take place.
* Central Park - The family often goes to the park for picnics and other activities.
* The Metropolitan Museum of Art - Toby and Peter visit the museum with their father.
* The American Museum of Natural History - Peter and Fudge visit the museum.
* Various stores and shops in the city - The family goes shopping for various items throughout the book.
While not explicitly stated, it's safe to assume the story is set in the present day, likely during the late 20th century, given the time period of the other books in the series.