When the mail server sends an "Over Quota" warning message it means your mailbox has or will soon exceed its default email account space limitation. The first notice is issued when an account reaches 90% of the space that is allocated to this mailbox.
If the issue is not addressed and 100% is reached, this could cause mail sending issues, and new mail addressed to you will be returned to the sender.
Ask yourself "Do I really need those 5000 emails? "Sometimes yes but web servers are not designed to hold massive amounts of email data - a much-preferred option if you want large data capacity is to use a dedicated email provider such as G-Suite, Office 365, or a Microsoft Exchange service.
It's the old story, you get what you pay for, and these providers do charge per account but once set up you'll have so much more data capacity - and dependent on your account level, this can be unlimited.
But if you are using a webserver to still collect your emails, the best way to avoid getting close to the quota is to monitor your mail on a daily basis, if you get an "Over Quota" warning, follow the suggestions below.
There are a few ways this can be done;
DELETE AUTOMATICALLY FROM THE SERVER WITH POP3
Set your email program to leave mail on the server for only a certain number of days. Log in to your webmail and remove delete mail from the sent items/inbox and then remove them from the deleted items or call your mail hosting provider and ask them to clear your mailbox. If you are using Outlook and collecting email using the POP3 Protocol, you can change your settings to delete emails from the server after a set number of days.
Use Webmail to access and clear your emails. Log in to your webmail and delete mail from the sent items or inbox and then remove them from the deleted items: In most cases use www.mydomainname.com.au/webmail and login with your full email address and password.
The second step in this process would be to remove the mail from the deleted items folder so that the space is freed up on your mailbox. Select Deleted Items and right-click on the Deleted Items folder, select Empty Folder. This will again bring up a message to confirm whether you would like to delete All the messages in this folder.
Once YES is clicked all messages will be removed from this folder and space will be freed up.
You should now have enough space to receive new emails.
In a number of email platforms eg Microsoft Outlook, you have the option to 'Archive' your email folders. Learn how to manage this and you can then continue to use your emails without pesky 'over quota' notifications'