While I agree that it sounds very much like a potential logic board issue, it's also possible that the power supply is having issues. If the PSU stopped providing power, the connector (or cable) that attaches to the Macbook is broken, or the jack on the Macbook broke (unlikely, but possible), the battery would have kept the system running until it ran down. Assuming this is the case, then there's only enough power left in the battery to activate the power light, but as soon as other components (such as the hard drive) attempt to start, it can't provide the power and shuts down again.
I'd check the power supply to see if it's putting out the voltage it's supposed to. Then, if you know anyone else with a Macbook Air, try theirs on your system.
I hope this helps!