iMac refuses to boot, Disk Utility always says there are drive errors it can't repair -- what to do?

Having solved the problem I can answer my own question.

It was a problem with the iMac's internal hard drive. Scanning the drive after the fact revealed a whole host of bad sector errors. Not sure why I couldn't pick this stuff up when I was trying to restore to it.

In any case, even with a new drive installed in the machine I couldn't get the Utilities -> Restore method of restoring the machine from my TimeMachine drive to work. It did the same thing with a new drive: it's just hang forever on boot, no chime, just Apple logo and a spinner for hours and hours.

The solution was to do a clean OS X install from the OS X DVD (instead of using the Utilities -> Restore menu option to restore the machine). Then, on first boot, when you first log in to OS X it'll ask you if you want to restore this machine from a TimeMachine drive. I said "Yes" to this option, it found my TimeMachine drive, I selected all the available types of things to restore (System Settings, User Data, Applications and Other) and let it do its thing for a few hours.

And this time it worked. Reboots were fine. My data is there and intact. All is well.

It sounds like this is a file system issue, rather than a drive issue, and you're right to consider restoring to an older backup. Unfortunately it's going to be hit-and-miss for a while, until you find a backup that works.

Once you're able to restore to a working version, I would do as DiskUtility says and backup your files and reformat the drive/reinstall the operating system, then make a brand new Time Machine backup from that.

Likely, the drive is fine, it's only something to do with a corrupt backup.