I think the biggest problem with hosting would be finding one that won't close your site no matter how much all the countries whose laws don't allow such content would press them to.

Also if you're going to start from scratch, no one would be interested in your empty site as long as ASSTR with its tons of stories maintains at least some semblance of functionality. You'll also have to advertise it a lot so that people at least know it exist, and advertising something that illegal is probably not that easy either.
On the other hand, if we're seriously concerned that ASSTR might shut down for good, it might be wise to make a complete backup of it and restore in case it really shuts down.

One more concern is that people are willing to put up with ASSTR permanently half-broken and totally outdated state because it's been there for ages and it is the only sourse with that many stories. But if you want to create a new site, people will expect it to be all modern and advanced with comments, personal messages, ratings, search by tag and all sort of stuff, and all of it actually working.

Also, as a side note, what makes ASSTR so special, besides the sheer quantity of stories, is how it has all kinds of stories there. There might sites dedicated to BDSM (bdsm library) or to vore (eka portal) or to loli (can't think of one on the top of my head) or to guro (gurochan, of course), and even if they allow other types of content, people would hesitate to post it there because it would feel off-topic, and the right categories just wouldn't be there to post it in. So if you're talking of making a new ASSTR, you have to keep this in mind, it should be all-encompassing.