I'm trying to get a 2006 Cobalt back on the road. It's presenting the symptoms of run down battery (Dim lights, time on the radio is reset to 12:00, won't start). The battery is original, so it's probably time for a new one anyway. The battery is not in the engine bay, though it looks like there are terminals to attach jumper cables to.
My Volvo has a similar setup in the engine compartment. It's battery is in the trunk. Try checking where the spare tire would be.
I found some helpful advice on changing the Cobalt's battery at yourcobalt.com.
...they continue on with step by step instructions.
The battery is in the trunk next to the spare tire under the carpet. You will need a 10mm socket for the bolts on the terminals and a 13mm socket for the bolt on the strap that holds it in place. A socket extension or deep well sockets would be handy.
Reverse the procedure to install a new battery. Note that you will lose your radio presets and possibly any current fault codes. I think the radio is tied to the vehicle VIN, so you should not have to worry about any lock codes or anti-theft procedures there.
There is a writeup with pictures at the yourcobalt.com forums. It says to disconnect the positive terminal first, but I was always taught 'Last on First Off' for the negative cable on batteries.
If you don't have a nice long socket extension, be careful not to short the terminals with your socket wrench.
I replaced the old battery with a Delco part from a Chevrolet dealer. If you go aftermarket, check for the presence and location of a hydrogen vent compatible with the vent tube. The vehicle will run without it, but I don't imagine hydrogen gas building up in the enclosed space of the trunk is a good thing. Remember, lead acid batteries give off hydrogen gas when charging.