You would need to log into cPanel and check if you have reached your actual disk qouta or if its just the mailbox qouta. If its the mailbox qouta, in cPanel under the mail section, in Email accounts, you will each mailbox has a qouta to it and how much they have used. The default qouta for mailbox's is 250MB. If the mailbox has reached that qouta you can just change it by clicking change qouta and give it the necesarry qouta or give it unlimited qouta. Having the mailbox qouta set to unlimited has a limit to it. The limit is the actual disk qouta of the hosting package the domain is on.
If the actaul qouta is full, you need to upgrade your package to the next bigger one. The last option and the one we hardly advise you to do is to delete mails of the server to save some space. If you are going to delete mails at a later stage, it would be best to set up your mails as POP3 accounts and run a daily mail backup on your PC. With Microsoft you are now able to store files and sync them daily from a cloud.