You have a failing internal hard drive, or some other hardware fault in the mass-storage subsystem. You must replace the drive, or otherwise determine the cause of the fault, immediately. But first you need to back up your data, if it's still possible.
There are several ways to back up a Mac that is unable to fully boot. You need an external hard drive to hold the backup data.
1. Boot from your recovery partition (10.7 or later), a local Time Machine backup volume (10.7.2 or later), or your installation disc (10.6.8 or earlier.) Launch Disk Utility and follow the instructions in the support article linked below, under “Instructions for backing up to an external hard disk via Disk Utility.”
2. If you have a working Mac, and both it and the non-working Mac have FireWire ports, boot the non-working Mac in target disk mode by holding down the key combination command-T at the startup chime. Connect the two Macs with a FireWire cable. The internal drive of the machine running in target mode will mount as an external drive on the other machine. Copy the data to another drive.
3. If the internal drive of the non-working Mac is user-replaceable, remove it and mount it in an external enclosure or drive dock. Use another Mac to copy the data.
I replaced the hard drive and still couldn't reinstall Lion, so I removed one RAM module at a time and on the second attempt, got Lion installed, so I think I had a failing drive and a bad RAM module.
I could not make an image of the volume with DU because it kept telling me the volume was corrupt.
It was time for an upgrade anyways. And I'll keep the old drive and bring it to a local shop for data recovery some time in the future.