Given the current infrastructure of the Internet, I do not believe that this is possible.
If it's stored online, anywhere, there will be some physical asset behind it that someone has some level of control over. Wherever that data is stored, someone will be able to disrupt it to some degree. Even if you were able to distribute the data in some way (like RAID5 for hard drives, except throughout many locations online instead), there has to be a way to address that data to retrieve it. An addressing system either needs a place to start looking (the DNS root servers, for example), or be constantly looking for peers that it can share data with and keep track of (think P2P networks). With a central place to start looking, someone can remove the glue and the record is gone (such as a domain name expiring). With P2P networks, it's generally slow or unreliable, and currently still relies on seeds to get started rather than blind connection attempts.
The other issue is one of control. If such a publishing system existed today, you would push data in that would not be able to be changed (think magazine publishing) because you would not be able to recall it (because nobody can take it down, remember), so it wouldn't be like a website with a CMS.
It almost sounds like you're trying to create some sort of "god" data that everyone can see if they look hard enough, but nobody has any control over to take away from those who have found a way to access it. I'm afraid this can only exist in the spiritual world. Anything we create we have the power to destroy. Everything in the digital world runs on physical hardware at some point, and that hardware can go away or be controlled.
Now for some theories...
A P2P network where the peers all talk to one another is the closest we've come to anything like this that I am aware of. The problem is that they still require seeds to get started, and someone has to host the seeds. If everyone ran a custom P2P software client which used a decentralized network of content directory servers which were automatically elected and rotated regularly, then you might start to come close if each "chunk" of content were distributed widely enough that it were always available. This would require some serious software and cooperation among pretty much everyone to run it to be feasible.
On a smaller scale, if you just wanted to hide one small website on the Internet, find a way to exploit the operating systems of all of the major enterprise level routers, then write a firmware update which includes a DNS server and a web server. It would inspect all packets flowing through the router and sniff out DNS requests for your site name, and if found, respond with an IP that the router would pass traffic for. Then it would be looking for requests on that IP and serve up the web page. This would be extremely difficult or impossible to implement from a practical standpoint, not to mention illegal and could be taken down when discovered, though it might be somewhat difficult to track if embedded in enough places.