It sounds like the website was build using an drag and drop website builder from iweb?
If they built the software used to create your website I'm afraid there's no way to transfer it to another host with this same software. However, you could always hire a web developer to re-create your website with software such as Wordpress (basically a free, popular & easy to use website builder). Once that's done it would simply be a matter of transferring your domain to a new company and opening a new web hosting account.
That would be my recommendation based on the situation. With Wordpress you'll be able to add/manage/change content on your website as you do now but without any risk of this happening again as Wordpress is free and supported by most web hosting companies.