How do you "get back without a backup"?
1. Use CarbonCopyCloner to create a cloned backup of what you have now. MAKE SURE that you set up CCC so that the target volume is HFS+. DO NOT USE the APFS option, or you won't be able to mount this drive later on.
2. Create a bootable USB installer with the Mavericks installer on it.
3. Boot from the USB installer
4. Open Disk Utility and ERASE the internal drive. NUKE IT BACK TO "ZERO" (Mac OS extended with journaling enabled)
5. Now open the Mavericks installer and install a completely fresh copy of Mavericks.
6. When done, run through setup and set yourself up with a user account.
THE SETUP ASSISTANT PROBABLY WON'T WORK. You can't use it to "migrate from newer to older", so you'll have to do a manual migration as follows.
7. Connect the cloned backup to MANUALLY MIGRATE stuff you pick-and-choose from it. (things like movies, music, pics, data, etc.)
8. You MUST select the drive icon and bring up "get info". Then click the lock icon and enter your password, then go to sharing and permissions and check "ignore ownership on this volume". Then close get info and you're ready to go.
9. You CANNOT COPY the folders named documents, movies, music, pictures (etc.) from the backup to your new account.
10. You CAN COPY "the stuff that's inside" these folders, however. So get your source and target folders visible in the finder, and copy the things you want to move.
Plan on at least a full day off to get this job done.
I suggest you keep HANDWRITTEN NOTES as you do the migration, to keep from going in circles.
You thought this was gonna be easy... eh...?