How Big Is My Inbox?

The app won’t download everything in your Outlook.com inbox to your phone. Because you use an IMAP connection with an app, you can access your e-mails without downloading the entire contents of your Outlook.com inbox onto your phone. When you set up your account on your mobile device, you’ll choose whether you only want to see messages from the past week, month etc… on that device. (don’t worry, you can still access older messages if you need to see them.)

Just to experiment, I added my Outlook.com  account to my Android e-mail app and despite the thousands of messages in my inbox, far less than a MB of data was added to the space the app takes up.
Now, if you are downloading a lot of attachments from e-mails, such as photos as documents, those can take up some space on your mobile device.

If, instead of just instant access to your messages, you want copies of all those messages to store – a mobile device is not the answer.  A PC would be better suited for that task. Find an e-mail client that will let you use a POP3 account and download copies of those messages to your PC.  The amount of space taken up by an average text e-mail is about 10 to 30 KB.  When you add attachments they’ll get larger – maybe even as large as 500 KB. So consider the number of messages you have and the type of messages you have and just multiply it out to take a guess at how much space they would use.
I don’t know of a way to determine exactly how much space your Outlook.com messages take up. But that’s normally not an issue, because there really aren’t any limitations on how many messages you can have.