It is an issue because unlike E-Mail Clients such as Microsoft Office Outlook or Thunderbird or Incredimail the messages don't download to the computer when using Outlook.com, Gmail, Yahoo Mail and other online applications.
I've found it slightly cumbersome but usable to double-click a message to get it in it's own window then use the Save As or Save Page As feature of a browser to save as a file, usually .htm/.html, on the computer's drive in a Folder.
The upside [if wanting to call it that] is that the messages remain on the service's Server until deleted and can be accessed from any computer you can use by just going to the site and logging in with either your E-Mail address or username and password. If using a 'foreign' computer don't allow it to save the password.