The story primarily takes place in Victorian London in the mid-19th century. This is consistent with most of Dickens' work, as he tended to use contemporary issues and settings in his writings.
However, the setting, other than the fact that it is often dismal, does not necessarily factor heavily into the work. Far more important than the physical setting is the "setting" of Scrooge. With Scrooge's greediness and lack of compassion as a backdrop, the story can easily unfold.
It is exactly this lack of backdrop integration that permits writers to rewrite the classic Scrooge tale into more modern settings. For example, the modern movie "Scrooged" translates (albeit loosely!) the ideas of the classic text into the world of a modern-day television executive.