2006 Chevy Cobalt undercarriage damage

It appears to be your gas tank. The straps holding it in place have rusted away and let it go. Judging from the extensive amount of rust showing on the unerside of your car in the pic, I would suspect being able to secure the tank in place again will be nearly impossible. I would say its time for a different car. A short term fix so you can at least move the car around would be to wrap a ratchet strap around the entire car, including the gas tank and tighten it up. That might lift the tank off the ground far enough that you can at least drive it home or get it to a shop for a better look.