As has been said, a larger companies are very unlikely to go belly up over night. If they do go out of business they will at least try to help the majority of their clients move on, they won't just switch off the servers one day.

Another thing you may want to do when signing up for a new host is find out how long they have been in business, if they opened last week then maybe you don't want to go with them if you are doing anything important.