I've made some changes that should cause the demo website to be available without having to republish it every few days.
I set up one of my other computers to run the ipfs client 24/7. I then added and published the website on that computer. The add should have by default "pinned" the files so they don't get gc'd.
Apparently the persistence on remote nodes would be caused by pinning files on those nodes. This would likely be handled by code written to support a decentralized cloud using an ipfs api.
I guess this is starting to make a bit more sense to me. Of course, if the files disappear again I'll have to figure out what it is I'm not understanding.