Hi Gary,
This is usually one of two things in Legacy Outlook for Mac. Either you are not opening the old Legacy profile, or the On My Computer area is hidden.
First make sure you are actually in Legacy Outlook. Turn off the New Outlook toggle at the top right and let Outlook relaunch.
Close Outlook completely. Open Outlook Profile Manager to point Outlook at your old local data. In Finder open Applications. Control click Microsoft Outlook, choose Show Package Contents, open Contents, then SharedSupport, then open Outlook Profile Manager. Look for a profile named Main Profile or any older profile with a larger size and recent modified date. Select it, click Set the default, then close Profile Manager and start Outlook again.
In Legacy Outlook open Outlook menu > Preferences > General and uncheck Hide On My Computer folders. Return to the sidebar and look for the On My Computer section. If your local folders were still present in that profile, they will reappear now. Once you see them, create a safety backup right away using File > Export and save an .olm file.
A quick note on what you saw earlier:
In New Outlook, Archive moves mail to the server's Archive folder, not to On My Computer, and Export can be unavailable, which is why it looked greyed out.
If after selecting the older profile and unhiding On My Computer you still do not see the folders, tell me whether you have a Time Machine backup. I will give you the exact restore steps to bring back the old Legacy profile from backup.