First figure out what kind of an email account it is: POP3, IMAP, or MSX (It's probably one of these three).
If it's MSX, 1) I'm sorry, and 2) I'm not sure how to help you.
If it's one of the other two, there are many email clients that support downloading the hierarchy of your email account onto your computer through their settings, but you will need to make sure you have enough space to hold all of everything before you begin, otherwise you're gonna have problems.
--Patrick