If you mean to store the addresses (as a file on your computer) then you have 2 options:
When you're on the Live Mail website, go to the task pane on the left hand side (on my computer, at any rate) and click "Contacts". Then select "Options" from the top right of the window; select "Export contacts" and save the file where you like.
If you're using Live messenger, then you go to the "Contacts" dropdown menu (Alt + C if you have Messenger set to look 'minimalist') and then select "Save instant messaging contacts" and do the same as for the website method. (I suspect this won't save those contacts which are purely via e-mail).