Here are five short stories (in no particular order, though without the usual Tolkien and Lovecraft), off the top of my head, that I recommend:
* "Undine" by Baron de la Motte-Fouqué
(a dark fairy tale, written in 1811)
Available here and here at Project Gutenberg.
* "The Machine Stops" by E.M. Forster
(a dystopian science fiction story, written in 1909)
Available here at Plexus.
* "The Willows" by Algernon Blackwood
(a weird/supernatural tale, written in 1907)
Available here at Project Gutenberg.
* "The Screwfly Solution" by Raccoona Sheldon (a.k.a. James Tiptree, Jr. and Alice Sheldon)
(a Nebula Award winning science fiction story, written in 1977)
Available here at SciFi.com.
* "A Study In Emerald" by Neil Gaiman
(a Hugo Award-winning story crossing the universes of H.P. Lovecraft's Cthulhu Mythos and Sir Arthur Conan Doyle's Sherlock Holmes universe, written in 2003)
Available Available here as a PDF at Neil Gaiman's site, and here as HTML
Your turn, if you so desire!
And, just for kicks, some of my favorite online text collections:
The Camelot Project
Discovering Sherlock Holmes from Stanford University
Index of The Works of Howard Phillips Lovecraft
The Literary Gothic
SciFiction (Classic and Original) at SciFi.com