A car engine would have no difficulty running for a week. It's the starting that kills them, as the oil isn't circulating immediately and the warmup is hard on them.
Also, nothing much shuts down during sleep. The brain is still functioning, the lungs are still functioning, the heart is still pumping, and for some people, the muscles are still moving. So what is really shutting down? The example of the dolphin just makes things even more perplexing. The dolphin does sleep half a brain at a time, but that's because it would drown if it shut down completely. So, why does it have to sleep at all? Whatever the gain is, it has to be pretty significant.

