The browser is a complete app environment except for one crucial piece: storage. It has a simple facility called localStorage, which almost fits the bill, comes close, but ultimately doesn't do what people want.
I have solved the problem in a generic and open source way. In a very popular server platform, Node.js. However it's not widely known that this problem has been solved.
Try this little app, as a demo: http://macwrite.org/.
You can sign in, write some text, save it, sign out.
And then sign in from a different machine, and voila, the text you entered is there. From that little bit of functionality you can build anything.
I have a new app in development, very simple, and brain-dead obvious, and useful, that builds on this. Hopefully at that point the lights will start to come on, oh shit, we're ready to build the next layer of the Internet. It really is that big a deal. And you don't need VC backing to participate. One developer, one person, can build something useful in a week. I've just done that myself. The service will virtually run itself at almost no cost, for a lot of users. That's an interesting place to be.