I can but concur with the other editors above - in this excellent short story we are plunged straight into the action as we are presented with the conflict that Della faces and the plot continues right until the sad/sweet end that really makes us question the act of giving gifts and what we have just witnessed. This, to me, makes this a good story - the fact that we still think about it after we have finished it.
Poe, one of the fathers of the American short story, said the two things a short story must do are be short enough to be read in one sitting and create a dominant mood. Both of those things are true of "The Gift of the Magi." The characters are clear, as is the plot--and the story starts, goes somewhere, and ends. Plus it has something to say. That makes it an effective short story for me.
About short story construction, the contemporary writer, Raymond Carver writes,
I think a little menace is fine to have in a story....There has to be tension, a sense that something is imminent, that certain things are in relentless motion; or else, most often, there simply won't be a story.
Certainly, O. Henry's "The Gift of the Magi" contains a little menace, albeit covered with a bit or wry humor from the intrusive narrator. It is a story of delightful tension.
The characteristics of a short story include that the piece is short, concise, and creates a single effect. "The Gift of the Magi" can be read in a single sitting. The story is concise in that all information provided to the reader relates directly to the main plot. After reading the story, the reader experiences the singular effect of irony. The story has a linear plot, a clear setting, characters, and a theme. All of these elements combine to create a masterpiece of a short story.
The story develops a strong theme about the nature of love through the development of the characters and the plot. Jim and Delia sacrifice their most prized possessions to buy Christmas gifts for each other. Irony is present, as well, in what they give up and what they receive from each other. Not every short story develops irony, but it is a common literary element in in stories. Also, many short stories culminate in a surprise ending; this one certainly does. (O. Henry was well known for his surprise endings.)
I agree that it has the qualities of a short story. You have a protagonist involved in conflict. You can definitely find the exposition, the complications, the crisis, the climax , and the falling action, along with the resolution. The story is also quite suspenseful.