Hmmm, this is a very tricky situation!! Usually it is difficult to change the hosting of an existing website without the help of the current host.
If your website is still live, and you are happy with a static HTML site (nothing dynamic like contact forms etc) you can sometimes do a "Site Grab" to copy the files off the internet. This way you'll have a copy of all your pages including html files and images that you could host on another server.
An easy way to do this is just to go to your website and click "File" > "Save As" and save the whole site page by page. Alternatively you could use a specialised program like WinHTTrack Website Copier.
As for the emails, that's a bit more complicated. You'll need to get control of the domain somehow, but the person who hosted your website and domain is probably the only person who is authorised. I would reccommend doing a WHOIS (just go to a website like www.register.com and search your domain. Then when it says "Taken" click "View details") to find out what company the domain is registered with, and contact them directly to see if they'll give you access to the domain.
Good luck!!
Tracey Munro, January 2009