Coolpix 8800 shuts down

Make sure to use good NiMH batteries.
Another possiblity is a problem with the card. Borrow a card and see if it has any effect. check to see that the pins in the card slot are all aligned, none out of place. check that the card door and the battery door close firmly, with some pressure. The camera can shut down when it has a problem detecting that the doors are closed.

When it does turn on, play with the zoom without taking pictures, zoom in and out, if during zoom the camera shuts down then you might have a lens problem, something is malfunctioning in the lens mechanism and the camera shuts down to prevent damage.

Good luck

Make sure the battery contacts are clean also the camera's battery contacts can get grubby too. You cold be suffering from an intermittent connection. Worth a try.