There are many web sites for online collections of literary fiction, many of which include short stories. Below is a small selection from what is available. Also, a Google search may turn up online versions of a short story (e.g., Faulkner and Barn Burning). However, as with all free web sites, it may be difficult to determine the accuracy and authenticity of the material.
Searching the catalog for short stories:
Short stories by English authors will be in the PR Library of Congress section, and short stories by American authors will be in the PS Library of Congress section. These will all be found on the 7th and 8th floors of Davis Library.