The story I heard (yeah, I didn't feel like looking it up either), is that the magic number 360 came from Mesopotamia. It was later adopted by both the Egyptians and the Greeks. The correspondence of the number of days/yr and the number of degrees/circle is because angular measure is intimately related with time measure: the year is the time it takes the sun to travel aound the celestial sphere due to the earth's revolution, and the day is the amount of time it takes the celestial sphere to travel around the earth due to it's rotation. The ancients didn't realize the causes, but they could measure the effect quite accurately.

As for the difference of the 5 1/4 days, that was either because it was difficult to measure these motions precisely with the tools available to them, so they got it wrong, or simply because they preferred the perfection of 360 to an odd number because it meshed with their number system. When they did notice the difference, some cultures simply added a long holiday to absorb the difference, just like they added leap years to handle the fraction. Incidentally, it's not exactly 1/4, which is why it is occasionally required to drop a leap day to keep the calendar in step with the seasons.

Something similar happens with the moon, too. The lunar month does not evenly divide into the year, either. The calendar month, which is roughly the length of the lunar period, is not exactly so, so the months kept on falling out of synch with our satellite, and eventually, the calendar police gave up even trying. By the time all these motions were thoroughly understood, in Greek and Roman times, the calendar was being modified for civil reasons, not astronomical ones.

The first calendar reform was done by Julius Caesar, and another was necessary over a millenia later by Pope Gregory. A further calendric reform has been proposed, but it is unlikely to be adopted due to religious and nationalistic objections (Hell, we can't even handle the metric system!). It might not be a bad idea, though, check it out.

http://en.wikipedia.org/wiki/World_Calendar

Personally, I think it should be adopted, but I also think all angular measure should be converted to radians, and all time measure made metric. I also believe we should abandon base 10 for our number system and switch to duodecimal (base 12), because 12 has so many divisors (2, 3, 4, 6), as opposed to 10 which only has 2 and 5. This makes converting fractions and dividing merchandise much easier (NOW you know why so many grocery items are sold by the dozen!) And of course, 12 pence to the shilling, twenty shillings to the pound...

Now for your homework, calculate the speed of light in furlongs per fortnight, and just how fast IS 3/5 of a mile in 10 seconds?