If you're using IMAP as the transport for your email accounts and have been receiving mailbox quota warnings then you'll need to move those emails stored on the serveronto your local computer.
IMAP basically keeps your emails stored on the server and your email program works like a window into those emails and allows you to use your email account as if those emails are stored on your local computer.
This is great if you use multiple devices and need to have say your phone and work computer showing exactly the same thing in your email software.
This is why 3rd party email providers such as gmail and office 365 have become so popular as they can provide gigabytes of storage space which shared web hosting environments can't usually provide.
For a detailed explanation on IMAP and POP please read this link
Click the Account Settings dropdown
Click Manage Profiles
Click the Data Files button
Give your data file friendly name such as your email address email backup e.g firstname.lastname@example.org_backup
Your new data file name will show up under the email accounts on the left side of the outlook window
Click the new data file name in the outlook email accounts pane
Create a new folder called inbox and an additional folder called sent items and any other additional folders that you have created previously under the new data file name
Now you'll move older emails that can be archived from your email address that's reaching its mailbox quota into the new data file stored on your local computer
This will move those emails off the server and onto your local computer, this will free up space for your mail quota
In the navigation pane in outlook
Click the inbox that is reaching it's quota on the server
Select all emails that are older than 3 months for example
Using your right mouse button click on one of the selected emails and click move
Choose Other folder from the resulting menu
Select the newly created Inbox folder that you made earlier
It may take some time for the move operation to complete if you're moving alot of emails from the server so be patient.
Once this has completed you can move the sent items folderand any custom folders into your new data file in the same way.