http://kb.mozillazine.org/Unable_to_process_the_backup_file_-_Firefox

jscher2000 has a tool on his website to extract out your bookmarks and save them to an HTML-format bookmark export file. If you want to try it, here is the page:

https://www.jeffersonscher.com/res/bookbackreader.html

To open the bookmarkbackups folder in Finder, use the "Open in Finder" button in the first table on the Troubleshooting Information page, and double-click into bookmarkbackups where you should see several dated files. From there you can drag and drop onto the conversion tool.

To import the HTML file, see the steps in this article: Import Bookmarks from an HTML file.

If you use Sync then best is to disconnect Sync temporarily during troubleshooting.

You can check for problems with the places.sqlite (bookmarks and history) and favicons.sqlite files in the Firefox profile folder.

• use "Verify Integrity" button on the "Help -> Troubleshooting Information" (about:support) page
  in case errors are reported close and restart Firefox and retry

See also:

• http://kb.mozillazine.org/Locked_or_damaged_places.sqlite
• https://support.mozilla.org/en-US/kb/Bookmarks+not+saved#w_fix-the-bookmarks-file

If "Verify Integrity" button on the "Help -> Troubleshooting Information" (about:support) page cannot repair places.sqlite then remove all places.sqlite files in the Firefox profile folder. Firefox will rebuild places.sqlite and restore the bookmarks from a recent JSON backup in the bookmarkbackups folder.

• keep a backup copy of places.sqlite in case a new places.sqlite database has to be created
• http://kb.mozillazine.org/Unable_to_process_the_backup_file_-_Firefox

You can use the button on the "Help -> Troubleshooting Information" (about:support) page to go to the current Firefox profile folder or use the about:profiles page.

• Help -> Troubleshooting Information -> Profile Directory:
  Windows: Show Folder; Linux: Open Directory; Mac: Show in Finder
• http://kb.mozillazine.org/Profile_folder_-_Firefox