It is important that you tidy up your mailbox regulary to avoid going over the limit of your disk quota. A good way to tidy up your mailbox is to archive your emails by creating a local folder which is stored only on your computer. Here are some guides on 3 popular email programs.
Thunderbird
In thunderbird, right click on the local folders and select new folder from the list. drag and drop emails into this folder. You can now view your emails off the server however this folder is only acessible on the computer you have set up the local folder.
Mac Mail
Go to mailbox, click on new mailbox, select the location to be stored on your mac, type a name for the folder, click ok. The new folder should now appear under 'On My Mac' with the folder name you have created.
Another good way to clean your emails on your mac is to Auto archive your email by adding rules , here is a useful link on how to do this: https://appletoolbox.com/2016/07/how-to-auto-archive-your-mail-on-macbook/
Outlook
Click the file tab, click on the accounts settings button. In the accounts settings window click on account settings and then click the data files tab. Next click add and in the file name box, type a name for the new local folder, click ok and now the folder will be created.