Sorry for the delay. There are several companies that can HOST your emails. But before I can help you find the BEST company for you, we must DEFINE what you want to accomplish first.
First, are you using emails with your own custom domain like @yourcompany.com? Or are you using free email accounts from AOL, Hotmail or Gmail?
Second, in some cases, when you HOST your emails, ALL OF OLD emails will NOT be migrated. Yes, they can stay on your devices but NOT in the NEW HOST.
Third, if you are using Gmail, you have the option to buy more storage or upgrade to Google Apps so that you can use your own domain name. Also, with Google Apps, you can migrate your OTHER emails to Google Apps so that you will have a central repository of your emails.
Can you please tell me which scenario describes your requirements?