Your options are a second macOS install on external storage and booting from that, then using that to pick off your files from the faulty install, then wiping the internal storage and re-installing, or (with a second Mac) using target disk mode and the appropriate cables and picking off the old data that way.
The former is probably the best path, and the external storage can then be reinitialized and re-purposed as a Time Machine backup once the initial file recovery has been completed. This is the same as erase-and-install, but with external storage.
Backups best happen before the disaster (and this is a disaster), and no backups means the data is considered worthless.