Peer-to-peer file sharing services like BitTorrent Sync are great ways to share large files without paying for third-party cloud storage, but they still require you to download software. FilePizza shares files using peer-to-peer right in your browser.
When you drag a file into FilePizza, you're given a link. Send that link to someone and they can start downloading the file right in their browser directly from you. Basically, your file never touches a third-party server, it's just a direct connection from your computer to the one receiving it. If you close the File Pizza window, that cuts off the transfer. The software's all open source too, so if you're worried what — if anything — it's storing, you can dig through the code.