The very first line of "The Boy Who Drew Cats" tells us exactly where it is set:
A long, long time ago, in a small country-village in Japan, there lived a poor farmer and his wife, who were very good people.
This is a Japanese fairytale, one of many translated by Lafcadio Hearn, an American writer who wrote extensively on Japanese culture, which at that time was largely unknown in the West.
Even without the assistance given to us by the opening line, there are a number of clues that tell us that the story isn't set in the West. For instance, we have the reference to the village temple, where the boy of the title is expected to train as an acolyte of the local priest.
By a process of elimination—Judaism, Christianity, and Islam don't have acolytes—this tells us that the people who live in this part of the world are followers of Shinto or Buddhism, the two main religions of Japan.
The big white screens on which the young boy paints pictures of cats are a common feature of Shinto and Buddhist temples. Shoji screens, as they're called, are made of paper, so they are perfect for the boy to paint on.
In years to come, the boy will become a famous artist, and some of his cat pictures will be shown to successive generations of tourists in Japan.