I remember seeing a documentary on TV a while back (which I can’t remember the title or channel) about sleep, or more important REM sleep. The program followed the experiments of a group of scientists who wanted to find out more about REM sleep.
Their experiment involved taking a group of students and letting them sleep. However, they woke them every 3 hours throughout the night. The students got a whole nights sleep, but because they were constantly woken from the REM stage of sleep, weird stuff started to happen to them. Although they did not feel tired in the normal sense of the word, they found that they could not concentrate on anything and would day dream and even hallucinate while wide-awake.
It seems that REM sleep is very important to keeping us sane and that we need approximately 4 hours REM sleep per night to allow our brains to function correctly while we are awake. By waking the students every 3 hours, they did not get enough REM sleep and so gradually started to act oddly.
This is why we require at least 5 hours of sleep per night as it takes about one to two hours to enter REM sleep after starting to sleep, and without REM sleep we'd simply go mad.
As well as the psychological benefits of sleep, we also need sleep to give our bodies time to relax and repair, however I know nothing about how much sleep is required for that, but I'd guess it depends how active you've been throughout the day.