The full version of Paperless has a feature to backup all of your data via email. (If you're using the free version, Paperless Lite, you can transfer your lists to Paperless instead.)
Backing up via email is an alternative to the automatic backup and syncing feature (use of the automatic backup and syncing feature is highly recommended).
Creating A Backup
To create a backup, go to the app settings screen and tap the "Backup To Email" button.
From there you'll be asked to enter the email address you'd like to send backups to. The email address you choose should be one that you can access with the "Mail" app on your iOS device.
You'll need to enter the email address twice, to ensure that you don't accidentally send the backup to the wrong person. (The next time you come to this screen to create a backup, the two email address fields will be pre-filled with what you entered.)
Next, press the "Create Backup" button. After a moment, an email composition screen will be displayed with the email containing your backup. Finally, press the "Send" button to email the backup to yourself.
Restoring Lists From An Email Backup
To restore lists from an email backup, open the previously sent backup email in the "Mail" app on your iOS device.
From there you can choose to either "Restore All Lists", or restore an individual list by tapping on its name.