Mavericks stuck on boot - spinning circle and loading bar, then shuts down?

Hi all - a friend of the family gave me their older iMac to look at. It's running Mavericks and the issue is that, once you turn it on...you get the Apple logo, the spinning icon and a loading bar. The loading bar proceeds to go across, then after a few minutes of being at the end, the whole iMac just shuts down.

I've already tried resetting the PRAM...already went into the Disk Utility and Verified the 'Base System' partition, which said it was fine.

The 'Verify' of the hard drive gave me an error something like 'Cannot do this to a Live Filesystem'

I tried doing a Repair on the disk anyway and it's still doing the same boot failure...spinning icon, loading bar goes across, then shuts down.

Safe Mode also doesn't work....started holding Shift after hitting the power button...same thing, loading bar, spinning circle, shuts down.

What else can I try? Is there a way to reinstall Mavericks, if need be, without affecting her passwords/files/programs so that those are all as they were?

Thanks for any help! :)