There is no date given in "The Gift of the Magi," so the exact time period in which the story is set is uncertain. The story was published in 1906.
Since Jim and Della live in an upstairs flat, we can assume they live in an urban location, although not in one of the wealthier areas. Della hopes Jim doesn't compare her to a "Coney Island chorus girl" when he sees her short hair, so they live in or close to New York City. Jim is employed, but has had a reduction in pay from $30 to $20 per week. It would appear Della is not employed.
Jim and Della live in a time when housing is difficult to obtain, food is purchased fresh from specialized markets, gas stoves are in use for cooking, and wages are low. Jim probably works long hours and has few benefits from his employment.