The storage limit in Outlook.com is 50GB. Currently, there is no way to see how much storage you've used.
If your inbox is full, you won't be able to send or receive any messages. Also, people who send you email will get an error message that your inbox is full. To make room in your inbox, try these solutions.
Empty your junk folder
Empty your Junk Email folder by right-clicking Junk Email > Delete all.
Sweep unwanted email from your inbox
Use the Sweep button to quickly remove unwanted email from your inbox or archive folders. Sweep lets you automatically delete all incoming email from a particular sender, keep only the latest message from that sender, or delete email that's older than 10 days.
To sweep your inbox, choose a message you want to sweep, and select Sweep.
To learn more about Sweep, see Organize your Inbox.
Save attachments to OneDrive
Saving photos and documents to OneDrive is another good way to make room in your inbox.
At the top of your message list, select Filter > Sort by > Size.
Note: If the message list sorts the smallest messages on top, click Filter > Sort by > Size again to put the largest messages on top.
Select the message containing the attachments your want to upload to OneDrive.
Under the attachment, select Save to OneDrive.