If you have a Full Access or a Web Access level mailbox, you are able to check your current mailbox quota size in the new Outlook Web App.
1. Log into Outlook Web App at https://na02.msexchangeoutlook.com/
2. In the upper right hand corner, click on the options wheel icon and select Options.
3. Under Account information you will see your current mailbox usage as well as what your current mailbox size or quota is currently.