MailStore Home is a free program that takes care of this. It can be used to backup all your email messages, no matter what e-mail program or e-mail service you are using. The program can also be used to migrate e-mail messages between applications and computers.
Click on Archive E-mail, and under "Create Profile", please enter your e-mail address of the service you are going to copy messages from, and then click Start. A new profile will then be created under "Saved Profiles" - double click on it to start the backup. Please note that the process can take up to several hours, if you have a large amount of messages, and if you are doing this for the first time.
The program also provides the opportunity search the entire e-mail archive once copied.
Should you use this program to back up e-mail from Gmail, then you will run into difficulties. Google, which owns Gmail, has in fact decided to block all third-party software from Gmail. Google has done this because it doesn't deem the program "secure".
Fortunately it is possible to fix this - visit the website support.google.com and click on the link "Allow less secure apps" and allow software that Google defines as "not secure."
Free Version of the program can be downloaded from mailstore.com - it supports Windows XP, Vista, Windows 7 and Windows 8.