For Webmail users who wnat to backup or archive mail, the only way to do this is to pull them from the server to a local computer using Outlook or Thunderbird etc.
Webmail is to view/interact with the email on the server only.
So if a user needs to back up messages they would need to set up a local access on Outlook or Thunderbird etc. and download the messages which would allow them to backup or archive as required.
From a safety standpoint it makes sense to store, especially important information, locally on your own computer as that way you have full control.
If you have an account that is to be closed and you need to save the existing webmail before hand, then all that needs to be done is set it up as a pop account on a local computer to retrieve all existing mail, then the information is safe locally. After this is complete you can safely delete the account.