FileTea is a free, web-based file sharing system that just works. It only requires a browser, and no user registration is needed. If you want to know more about it, you can read my previous blog post. For a more detailed description, read Nathan Willis’s excellent article on LWN.net. There have been a few changes since that article (HTTPS support in particular) but it’s still the best one you can find on the net.
Igalia still provides a FileTea server at http://filetea.me/, that you can use to share your files and see how it works. We plan to keep offering this service, but you don’t need to trust it/depend on it anymore: now you can apt-get install filetea and have your own.