It already exists - of sorts. It's called Freenet. It's not an easy place to navigate, nor pleasant with regard to content, and the fact that it is ridiculously slow doesn't help either. It's a completely anonymous network which is made up by the clients itself, think the Internet powered by peer to peer. All files are stored and replicated across the network and then given a hash number. Web pages and files are then recovered through the use of these hash numbers. So in order to download files, you need the hash, your client then asks other connected nodes if they have the file and/or if they are connected to other nodes with the file, and it propagates through the system.
The thing is, you need to leave the client running 24/7 to make use of Freenet, and it takes about a day before you can actually do anything. It's also a bandwidth hog and needs about 10GB of local storage to be even remotely effective. But the entire system is built up of clients and a few dedicated nodes. The other problem is that it died out back in late 2008, so the network is filled with dead files and a lot of junk and broken links, it can take something like an hour to download a page. But the more people that use it, the faster the system becomes. Think Tor but with data storage as well.
Usenet is like a subnet of the Internet, and Freenet is like another subnet... which in-turn has another subnet called Darknet. Darknet is Freenet but with limited, user specific nodes, think of it as a friends list, so you and your 'friends' build a separate dedicated network built on Freenet. Needless to say the shady nature of the whole system, and yes, it does get used for such purposes.
The key point to the whole Freenet system is being able to remain 100% anonymous, and the developers go to great lengths to keep it that way. It's been banned in France I believe, but as always, there are ways around it. It was used to leak government documents, certain anti-piracy company emails and documents, like Media Defender. All sorts of things.
Very steep learning curve to use, you need about 3 or 4 applications, 10GB of disk space, one week and a tonne of bandwidth to use the system. After that, good luck trying to find anything, it's like walking into a teenagers bedroom...