The internet has proved to be a great resource for finding fiction no longer under copyright restrictions. Here a few sites that maintain such public domain works:
Free SF Reader: A large, well organized collection of free science fiction, fantasy and horror.
Free Speculative Fiction Online: A list of links to free stories online.
Page by Page Books: A listing of many free classic books, including Verne, HG Welles, and Edgar Rice Burroughs.
Project Gutenberg: A vast collection of public domain texts, searchable by author or title.
Novels and Excerpts
Baen Free Library: Many free science fiction and fantasy novels from Baen publishing. One of my favorite reading sources!
Del Rey Science Fiction & Fantasy: Novel excerpts from long time genre publisher Del Rey.
Infinty Plus: This site is full of free novel excerpts, short stories and non-fiction in the science fiction, fantasy and horror genres.
Online Short Stories
Tad Williams' Shadowmarch: Some short fiction relating to the author's Shawdowmarch novels
Tor.com Stories: Reprint and excerpts from one of the genre's leading publishers.