In File Stream, you can select which content is "made available offline", which means "copied to a local directory on your computer and kept in sync with any changes made to the online content by you or others". So, if you want to back this up to a NAS, you need to select all content and select the "make available offline" option. Since you don't want this for all users, I would dedicate one computer for this task. I don't know what happens for new folders created at the top level, so I would test this out so you know. Maybe put all content under a single top-level folder and then set that single folder as available offline on your backup computer?
This is really no different than the "Backup and Sync" product for the personal version of Google Drive.