Hey, I'm just going to post what I know, and I have never tried actually downloading an entire mailbox from Exchange Online.
To create a ZIP file of the contents of the users OneDrive, you can select the User in the Admin Center, go to the OneDrive tab, and then the first thing there is a link to their OneDrive. Select all files, click Download.
For Exchange I'm not sure, but I know you can search mailboxes and export the found items as a PST. I just tested it, and if you run a search with an empty match/query, it seems to return all items in the mailbox.
You can do that in the Compliance Admin Center under "Content Search". On a new search, enable searching Exchange mailboxes, and then next to that select the user. When the search is done, open it and go to "Actions" -> Export, it will eventually show up as a finished export on the Export tab in Content Search, and it will download a little tool that ultimately saves the PST file to disk. I have found this only works if I do it through MS Edge.
Now I see you can potentially also do a content search on OneDrive, but I don't know what the results of that would be.
