My nephew David turned me on to Flickr. It works quite nicely for storing, navigating, viewing, and tagging pictures. Flickr is well worth a look.
Flickr only does pictures, so I went hunting for a decent place to store other media types. I was primarily interested in finding liberal storage, minimal upload constraints, a zippy web UI, the ability to feed off rss for pictures from other folks, and a promising API for future projects. Oops - forgot to mention one more critical feature: no cost.
I've found two that look particularly promising. They don't meet all my criteria, but they seem workable enough - we'll see...
Google Video provides storage with a very liberal policy on size and upload frequency (i.e. no real limits). On the minus side, they use a flash player to display videos, making it hard to use them as a storage mechanism.
Ourmedia is another promising candidate. Like Google Video, they are rather liberal in their upload constraints. They use Internet Archive for storage, so I imagine disk space is not a problem. My primary gripe with the site is the painfully slow performance. On the plus side, they also support audio, pictures, and text uploads. They also avoid the use of flash for viewing the videos, so you aren't locked into their web UI.