Re: (broken) iMac hard disk backup
If it for sure can't start up (I suppose you have already tried to use CD/USB to boot) the only way is to take the HDD out.
If it is a 24'' iMac, that means it is from 2009 or earlier, which is really easy to open up, you just need a suction cup and some torx screwdrivers. Check ifixit for detailed guides.
Since you only care about getting the data out, you have nothing to lose. I would help you out for fun, but I live too far away.