Yes. Boot recovery mode and determine if you can restore a "Local Snapshot" of the previous installation:
- Boot macOS Recovery.
- At the Mac OS Utilities screen, select "Restore From Time Machine Backup" then Continue. A Restore from Time Machine screen will appear. Read it then Continue.
- When the Select a Restore Source screen appears, select the internal disk. Its default name is "Macintosh HD" but you may have changed it. Continue.
- The Select a Local Snapshot screen appears next. Restore From > Local Snapshots on Macintosh HD (or whatever you named its startup disk) will appear in the dropdown menu selection.
- If a Local Snapshot Date & Time list populates, select the most recent one. Continue.
- A confirmation dialog will appear. Read it then Continue.
- A Restoring screen will appear next. That operation won't take long.
- A Success screen will appear next, and the Mac will restart.
If Step 4 above is not an option for you (due to never having turned on Time Machine), you can select Terminal from the Utilities menu and possibly proceed to delete files.
Another thought is to boot Safe Mode which first performs a cursory directory verification and repair, and it also deletes certain cache files maintained by the system, which might recover just enough space to proceed with the installation.
If that doesn't work then attempt to reinstall macOS using that option from the macOS Utilities screen.
If all else fails... the good news is that your files appear to be present and intact, so the only obstacle to overcome is to boot that Mac from some other device. You may need to purchase an external hard disk drive and install macOS on it, using the same reinstall macOS procedure. To install macOS on that disk select it in the Show All Disks button that appears. You'll just need to format it first using Disk Utility.
Then, when you eventually recover from the shock of having almost lost everything on that Mac, by all means use that new external hard disk drive for Time Machine and start using it.
