There is not a limit set for individual email accounts other than the capacity of your hosting account's storage space limit. In order to avoid reaching or exceeding your hosting account's storage limit, you can define the mailbox storage quota on any of your mail accounts via your control panel interface. If set, once the mailbox storage quota is reach, you will receive warnings to let you know you should clear out old emails.
This tutorial will assume that you are logged into your cPanel. If you are having trouble doing that, please see the tutorial named "How to Login to cPanel".
Scroll down your main cPanel page to the section called Mail, and click on the icon that says Email Accounts. You will see a list of the email accounts you have created.
>>Click Change Quota next to the email address for which you wish to adjust the quota.
>>Type in your adjustments and click the change quota button
You can adjust these limits as needed and as many times as you need to. If you give it a set limit of a certain number of MB, and the box becomes full, you will get warnings to clean it out. You may leave your mailboxes set to unlimited, but if you don't remember to check and empty old mail off the server, you run the risk of using all your server space.
When you are done, click on the home icon in the upper left corner to return to your cPanel. This is the end of the tutorial. As always, if using a public computer, remember to close your cPanel by clicking on Logout rather than just closing the browser window.