There is a half-way option.
You buy yourself some web space. eg. You register your own Domain. You rent web-space on which to build your site.
You can then download the Forum software, such as phpBB from phpBB.com (the software we use).
And off you go. You need a web-site provider that allows FTP access, and can run mySQL, but all those technical specs ar at phpBB.com.
You have full control over how your Forum looks, and the data. Which you can pick up and move to another Provider whenever you like.