[[ I have a 24" white iMac. I have run some test and states the hard drive is failing. I used disk warrior to try and repair to no avail. Instead of me taking apart and replacing. Just need to know if running a fresh install on ext drive would be suffice? ]]
The iMac will boot and run from either a USB or a firewire external drive.
Boot times WILL be a little slower -- no way around that. But -- more importantly -- once it's "up and running", you probably won't notice much difference (if any) from an internally-mounted drive.
If you have a firewire 800 port on the rear, a drive mounted in an external case that has firewire 800 would probably be the best way to go. It will probably yield boot speeds 50-60% faster (or more) than you would see from USB2.
I boot and run my own 2007 24" white iMac from a USB2/SATA docking station with a bare drive inside. Boot times are slower, but no problems when running.
Be aware that an enclosure with firewire 800 will cost a little more than one with just USB2 (or USB3).
I think the best reason for choosing an "external solution" for an older iMac is that it's not really worth spending the money to fix a computer that you are likely to replace in another year or two. You could do as well to buy something "up to date" such as a Mac Mini, and be far ahead speed-wise. And you could take the external drive "right over to" the new Mac, very easily...