Down to two possible problems

[slader99]

Disregard the post below, I didn't notice that you had mentioned the wire with burns went to the frame. No problem with having an extra beefy wire running to ground!

Where was the "burned piece of wire"? Take a picture of its location and post it for us. Insulation missing on a ground wire wouldn't cause you any issues whatsoever. As long as the ground conductors aren't severed, there is no need for insualtion on them.

It sounds to me like its a hot (live wire) that is shorting against your body. That would explain the unwillingness to start even with jumpers attached.

If I were you, I would jumper the car again, and while the ignition switch is in the on position, use your voltmeter to see if there is voltage between that burned wire and the frame of your car. If you want you can skip that step and jump straight to replacing the section of wire with a piece of similar gauge. (You don't want a piece of 26 gauge wire with a 8 gauge section in it, it looks crappy and will be hard to join effectively) If it is indeed a live wire that has been shorting against your body, your battery is probably now pooched because automotive batteries don't take depleting to emtpy very well.....

If that burned piece of wire is your problem, after you've patched up the wire nicely (I'd solder then heat shirnk or electrical tape your joints) the car should have no issue starting when jumpered to a heathly battery.