Yep, using the export
works (I've successfully exported quite a lot of pages). Although that form allows adding page names via the categories you can also simply add the page titles into the edit box. Following the instructions here
special page you can get them fairly easily and I've confirmed I can export fairly large chunks of the Wiki to XML. It'll take a few goes to get the "(Main)" namespace as it seems there are so many pages there that it won't display all of the links at once. I doubt there is much value to archiving the User/Talk namespaces etc, but for completeness it's probably worthwhile and isn't that much extra hassle as there aren't that many pages there.
File attachments appear to be the most problematic bit - however there is some python+perl to grab them here
- although no solid advice as to how to easily get them all re-uploaded without installing an extension. Spending a couple of hours writing another perl/python/whatever script could probably automate the upload too I'd guess - I'm surprised they've not already got that.
MediaWiki do seem to have a list of free wiki hosts, although I cannot vouch for any of them. Do people like github/sourceforge support it too? Probably?