You can confirm this by logging into your Customer Control Panel > click the Mail Tab, look under the Storage Size column.
To empty your mailbox you have a number of options.
Option 1 - Download your mail with POP3 using a mail program such as Outlook. See: How to configure Outlook using POP3
Option 2 - Within the Customer Control Panel > Click the Mail Tab > Click the Display Name of the user in question > Click the Forwarding Tab, Click Forward List > Click Add Forwarding E-mail Addresses, Submit changes > Click Storage Tab > Click Delete.
This will delete all storage for the mailbox in question. Note: You will need to Add Storage again in order to receive mail again to your mailbox.
You may also want to disable forwarding.
If you do NOT want to empty your mailbox, instead increase your mailbox storage size you have two options.
Option 1 - Upgrade your shared hosting subscription within the customer control panel via Account Tab > Click Subscriptions > Click Switch Plan to choose a different subscription that offers more resources for mailbox storage
Option 2 - Upgrade your mailbox to Hosted Exchange which gives you 2GB of mailbox storage by default. You can purchase a Hosted Exchange subscription within the customer control panel via Account Tab > Click Buy More Services > Click Hosted Microsoft Exchange, follow the on-screen options to complete checkout.