Heres the thing... Your E-mails are not on your computer hard drive. They are on the server on the company that is your provider. You cant go directly to your computer to access them. You have to get on line, access the server.
Here in Tampa we are having lost e-mail service with Road Runner,Brighthouse. Brighthouse said dont worry,all e-mails are safely stored on their servers, and eveything is backed up.