Email quotas are generally used to keep mailboxes to a reasonable size.  Beyond improved manageability, smaller mailboxes cost less to maintain, especially when "pay by the byte" offsite backups are involved.  Here are some very quick things you can do to "take out the trash" and reduce the size of your mailbox:

First, find and delete the largest unneeded items.  Here's how:

1) In your inbox, click the search field at the top.  The search ribbon will appear at the top of Outlook.

2) In the ribbon, click "All Outlook Items", and then "Has Attachments".  Email results will appear.

3) In those search results, switch sorting to "Size" -- Right-click on the existing headers and select "Size" from the list.

4) Then sort by the Size column to sort the largest emails to the top, and delete 10 or so of the largest items.  
Note: If they have needed attachments, just right-click and save those to your local computer.

Finally, empty your Deleted Items folder.  Here's how:
* Just right-click on 
Deleted Items in your folder list, then select Empty Folder

These two actions should take 5 minutes or less, and make a large impact!