Running Craft version 2.3, build 2644, schema version 2.3.4... we clicked "update" and got an error screen that looks like the install screen that simply says "oops".
How can I restore my site from the files that are in the (automatically generated) folder "app.bak"?
You can restore your database from your pre-update attempt database backup that you (hopefully) made. Assuming you've got the the backupDbOnUpdate config setting set to true, Craft made one for you in craft/storage/runtime/backups as well.
You can also download any previous version of Craft's files via: Where can I download previous versions of Craft?
Just replace the existing craft/app folder with the fresh one you downloaded and delete app.bak. I wouldn't recommend restoring directly from app.bak because depending on where your update failed, it could be in an inconsistent state.
I'd recommend doing a manual update when you're ready to update again as you probably ran into your server's max execution time or memory limit (which tends to happen on these larger updates).