The lack of a startup disk in the panel suggests the boot loader on the boot disk is missing/damaged etc.
Etresoft has covered how to backup in Disk Utility. That will allow you to get a copy of the data (assuming the disk is still in a mountable state).
You may want to verify the volume is not damaged whist in recovery mode - select the indented 'volume' for your boot disk & verify that - repair if required (the disk is the item above that). Use the restore tab for the volume to back it up.
If you cannot afford to erase the destination disk you can save a backup to a disk image - this is less flexible, but it will have al the data in one disk image. The method involves using Disk Utilities 'File > New disk image from folder…' then select the base of the boot disk as the source. Save the final disk image to the external disk. Don't use 'New disk image from volume', that option saves the 'full size' of the volume including empty space!
Note: a full backup copy can be used with Migration Assistant whereas a disk image backup cannot, but you use manual methods (or apps) to restore from the disk image.
Etresoft is also correct about upgrading - wait until this OS boots correctly.
Reinstall is the default option in the recovery menu - that does not delete your personal data, it copies the new OS files over the top of the old ones. That should replace the damaged/ missing boot loader (or whatever else is broken in the OS).