You will need to delete unwanted emails from your mailbox to enable you to send or receive emails, also if you reach your mailbox you will be unable to send/receive further mails.
You can delete emails from your mailbox by:
- Logging in to your mailbox and delete unwanted emails
- Remove emails from the server by using email software
Alternatively, upgrade your organisation to Microsoft Office 365 Small Business (which includes 50GB mailbox storage per licence).