I had a very similar idea yesterday when I was thinking about other possible applications of BitCoin's distributed/anonymous/proof-of-work model, and I'm sure we aren't the only ones. As a technical challenge or experiment, I think it's great and certainly interesting, but in terms of use cases it's quite limited. I use Gmail today. It's secure, well-designed, free, in the cloud and I don't have to run any kind of software on my computer to make it work. The only drawback is that they scan my emails to show ads, but that's a price I'm willing to pay for the convenience, and a price most people are willing to pay.
And why not take it one step further and extend it to rethinking the whole model of the web? Why do we have domain names? Let's BitCoin-ify the way we connect and interact with remote hosts, and not just leave it at email.