The only possible element that could be left out without affecting the quality and integrity of the story would be exposition. If nothing has taken place before the story starts, and therefor the author does not need to explain anything, exposition would not be necessary. I cannot think of a story like that at this moment. None of the other elements could be left out. Consider the function of these elements in a short story.
exposition-The exposition explains events that have taken place before the story starts and which the audience will need to know in order to understand the story. If nothing like this has occurred, the exposition element could be left out.
conflict-The conflict is the problem that sets the whole story in motion. Without a conflict there is really nothing to build the interest of the story.
rising action-The rising action are events that take place in the story building the character involvement and intensity, but also building the interest of the reader in the story. Without this, a reader would probably not be motivated to read the story.
climax-The climax is the highest peak of intensity in the story, and it paves the way for both the falling action and the resolution. Without this, there is really no place for the story to go.
falling action-The falling action brings down the intensity for both the character and the reader. It winds the story to a resolution. Without this, there would be no way to reach a resolution, and the reader would be stuck at the high level of emotional intensity.
resolution-The resolution resolves the conflict. Without this, the reader would be left with a feeling of incompleteness.
Clearly, 5 of the 6 elements are essential to a quality story with a satisfying reader experience.
eNotes has an interesting link on the development of the short story as a genre.