When managing emails in Outlook, the way your mailbox interacts with the server depends on whether you're using IMAP or POP3.
With POP3, emails are downloaded to your device and often removed from the server, freeing up space. IMAP, on the other hand, synchronizes emails across devices while keeping them stored on the server, which may eventually lead to a larger or full mailbox.

If you prefer to keep all emails, the best long-term solution is to request that your administrator increase your email storage space. This allows all emails to be stored on the server, where they can be accessed from any device connected to your account.
Additionally, storing emails on the server reduces the risk of losing important communications due to hardware failure, theft, or damage to your local device.

However, if upgrading storage is not an immediate option, here are some steps to free up your mailbox space:

In Outlook, you can archive and store old emails locally on your device by moving them to a separate Outlook Data File (.pst). This file can be accessed in Outlook whenever needed.
To do so, please refer to this guide: Archive emails manually in Outlook

Note: Once archived, these emails are removed from the server and stored on your local computer.

Clean up your mailbox by deleting emails you no longer need. Focus on large emails or those with attachments.

Attachments can take up significant portion of your mailbox storage.
In Outlook, you can remove attachments from an email to free up space.
Note: Ensure you save a copy of the attachment locally before removing it, as it will also be deleted from the server.

Apart from manually archiving emails, you can also set up auto-archiving in Outlook to automatically move older emails to a local archive on a regular basis.
To do so, please see: Archive older items in Outlook automatically