I have a backup of the entire Wordpress folder of one of my old websites but, unfortunately, I forgot to also make a backup of the database for it. It's been years since the website expired so there's no way I can get the SQL file that way, and both Wayback Machine and Google don't seem to have any caches left of it (a tutorial I saw mentioned trying that way).
Is it still possible for me to get the website working for local use?
Yes and no, depending on what you are willing to do.
Is it still possible for me to get the website working for local use? No
If you have no database you will not have access to the Posts, Pages, Users, Options and all of the meta.
Is it still possible for me to get the website working for local use? Yes
If you create all of the database entries again to your satisfaction. You will still have all of the images you can re-upload to recreate the database entries for images. Then just re-create all of the pages and users as needed. Plugins may need to be disabled/reactivated/reinstalled and configured as well as your theme settings.