Usenet is a public service that has been in use since the early 80s. It is probably the oldest form of sharing on the internet. There are groups of similar interests and people can freely discuss in these groups.
Technically there are quite a few differences of course, since usenet uses its own protocol, and is not really a centralized cloud, but more a mesh of individual servers that share data across. However, the way usenet works from a user perspective is pretty similar to Google Wave.
Second (if not first) largest use currently for usenet is binaries. From warez (illegal software copies) to full DVD or Blu-ray images, and console game rips. A lot of copyright organizations are trying to crack down on usenet and ban it all together. In the Netherlands BREIN has forbidden sharing of links to nzb files (a collection of "links" to the actual contents on usenet) on usenet sharing sites.
How does Google Wave fit in all this? Well, Google Wave also allows sharing of documents and files. With its closed nature people can start "elite" groups to share information like warez and rips. People could freely share nzb files or links without anyone knowing outside of that "elite" group. Google Wave also supports public groups, where everyone can join. That could be the new place to share warez and rips too. I don't know how large the files in Google Wave can be, but it is easy to break a 5 gig DVD rip into smaller parts and submit that into a wave (just like usenet posters do, usually posting many smaller files instead of one big file). Downloading would currently be a little more cumbersome than usenet, but don't forget that the Google Wave API will probably allow downloading of multiple files in one go in the not too far future.
Google has certain user agreements that sharing of illegal content is not allowed, but do they have the power to control once Wave has gone public and probably millions of users are using it? Especially if it is going to used as an "elite" like group share using closed Waves. And the terms of service essentially say that you are using Google Wave "at your own risk".
Ah well, for now Google Wave is just a nice toy, we'll see how far it goes once it is really public.