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_USDiskWarrior 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/HT201475Once 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.comAnother 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/