What year was the story A Christmas Carol set in?

The story was published in 1843, and takes place in the early 1800’s (1800-1850).

Dickens is careful not to give a specific year that the story is set during the story.  This adds to its timelessness.  The closest he gets is the comment that the Ghost of Christmas Present makes when Scrooge asks about his family.

“Have never walked forth with the younger members of my family; meaning (for I am very young) my elder brothers born in these later years?” pursued the Phantom.

“I don’t think I have,” said Scrooge. “I am afraid I have not. Have you had many brothers, Spirit?”

“More than eighteen hundred,” said the Ghost. (Stave 3)

This means that the story takes place at least after 1800, of course.  It likely that the "present" is set approximately around the year it was written.  The ghost is telling Scrooge that he lives for only one day (which is why he says he is young), and when he says that he has more than eighteen hundred brothers and sisters he means that each of them represents Christmas Day.  Scrooge’s response is really more of a quip, but he is, as always, thinking about money.

Of course, while 1843 might be Scrooge’s present, the story itself shifts into his past and his future.  Thus, it takes place anywhere from the beginning of the nineteenth century to the mid to late-1850’s.  It does not take place in just one year, because Scrooge and his ghosts travel through time in a way.  However, they still end up back in Scrooge’s present.

Dickens intended for his story to be read for years and years.  He wanted to make it a timeless classic.  In that sense, he succeeded admirably.  People have still read and enjoyed this book for years, and continue to do so today.