To store your data on a local PC, or other medium, you can use a local data file (PST).
You can create a local data file (PST).
You create it by choosing "New items > More items > Outlook data file" from the menu.

Outlook will then ask where you want to save this file and what name you want to give it.

Once you have done this, an additional item will appear in outlook which may look like this:

But if desired, create an additional folder so you can easily see what mail is in this file:

You can now save emails in here which will then no longer take up space online. Make sure you make a backup of this "just in case".