Well you could print them out, and scan them put them on a disk that way.
Web based email, such as Yahoo, Gmail and Hotmail, you "read" a copy of the email on their servers. A server is basically a super fast and larger volume computer, most suited for network traffic. Most web based emails you need a web browswer (IE8 or Firefox are the two biggest).
Emails from ISP's (such as AT&T, Comcast or Time Warner), the copy is sent to you directly. You however need an email application, such as Microsoft Outlook. There are other free email managers, Thunderbird i belive is a popular one. This is normally called "POP" email, which i belive stands for Point Of Presence.
Yahoo has Email Plus , if you subscribe to them and pay an annual fee, this normally will allow you to access your emails without a web browser.
try this out http://overview.mail.yahoo.com/enhancements/mailplus