Mountain Lion now supports multiple target volumes to allow you to rotate backups - but you still can't use it to backup over the internet.
Integration with the system means that some Apple apps, including Mail, iPhoto and GarageBand, can recover data from Time Machine directly.
Use Time Machine for backups - as long as you don't want them to be bootable
Use Disk Utility
Apple's Disk Utility is one powerful tool. Aside from formatting and partitioning hard drives, creating disk images, mounting disks and checking a disk's S.M.A.R.T. status, it can be used to solve a variety of disk problems. Best of all it's free and sitting in your Utilities folder.
It's a good idea to use Disk Utility after a crash or freeze. If you start up from your boot drive, you can only verify the volume. If you find a problem, you'll need to boot your Mac via a different method. Prior to Lion, this would have been an external bootable drive, the original DVD or a retail copy of Mac OS X. But OS X Recovery changed all that.
When you install Lion or Mountain Lion, the installer creates an invisible, bootable partition on your boot drive. This contains Disk Utility and a few other essentials including the ability to restore from a Time Machine backup and to reinstall Mac OS X.
Of course if the problem is a mechanical one with your hard drive, this partition probably won't help much. But for the sake of holding down Command-R on startup, it's worth a try.
As for repairing permissions, it's unlikely to help much.
Use Disk Utility to verify your hard drive and repair certain problems.