These steps change quite often, fair warning!

Steps below are as of 2021-01-04. Please note this has to be done in Internet Explorer or Edge.

First, set permissions.

1. Browse to protection.office.com/homepage , log in as tenant administrator
2. Click eDiscovery Manager.
3. Open eDiscovery Administrator, and add your current admin user.

It takes one hour (as of Microsoft support 2021-01-04) for the permissions to take effect. Used to be up to 24. Afterwards, perform the export.

4. Browse to protection.office.com/homepage , log in as tenant administrator
5. In left pane, click Search, then click Content Search
6. Click New Search
7. At bottom, item Specific locations, click Modify
8. At top left, Exchange email, click “Choose users, groups, or teams”
9. Click “Choose users, groups, or teams” in the new window
10. Enter mailbox email address, wait until the search results come
11. Check the mailbox, and click Choose, then Done
12. Click Save
13. Click Save & run
14. Name the query (and PST), click Save. The query will run. Don’t continue until it’s done. It will say “Status: completed” at the lower left.
15. Click on the word “More” at the top, just to the left of the word “Sort”.
16. Click on “Export results”. Choose options appropriately. Click Export.
17. Click on “Exports” above that area, to the right of “Searches”. If the export does not appear, click on Refresh.
18. Click on the export item.
19. Click on “Download results”. A download applet will start, requiring an export key which can be copied from the browser. Paste it in, choose your download destination, and go! It can take a very long time to start, there is a long preparation phase.