hi there, almost everything is possible given enough time and money If I understand your request, you need to migrate a site from WordPress to Joomla, and also change the hosting company. Is that correct? These are two separate issues. Changing from hostgator to DreamHost is the easy part, there is plenty of articles on help.dreamhost.com you can ready to learn how to move hosting.
The migration from WordPress to Joomla is a whole other piece of work. In theory it can be done, in practice there is no magic wand or a simple recipe and you'll have to learn how to do that yourself.
hope this help