If you have access to another Mac, and a Firewire or Thunderbolt connection between them, this will be useful: https://support.apple.com/kb/PH10725?locale=en_US

DiskWarrior as Tony mentioned is pretty much the goto tool for dealing with HFS corruption that Apple's own Disk Utility won't fix: http://www.alsoft.com/DiskWarrior/

You could run DiskWarrior on the second Mac with the iMac in Target Disk Mode. Or as Tony indicated, booting the iMac off a DiskWarrior USB stick.

As for reinstalling once data is recovered, the 2010 iMac supports Internet recovery, no need for discs: https://support.apple.com/en-us/HT201314 . This process may install an older version of OS X, this will lead you through the steps to get it back to El Capitan: https://support.apple.com/en-us/HT201475

Once back to full health, definitely get them to use Time Machine to take care of the onsite backups. For offsite, I recommend Backblaze at https://www.backblaze.com

Another approach to offsite if it's mostly their photos they want extra protection for, iCloud Photo Library is a good way to go. This would also sync and merge all their photos between the Mac and any other Apple products such as an iPad. https://www.apple.com/icloud/photos/