I am not an expert at this, but I will take a SEWAG at it (Somewhat Educated Wild Ass Guess).
First, 360 degrees is not a metric thing. Both English and metric measure use it. And there are other units for measuring angles: 400 Grads, 2 pi radians, mils (the angle that subtends 1 m at 1 Km), NATO mill, milliradians, etc.
Early philosophers (that's scientists in today's lingo) sometimes had a need to measure short internals of time. But the hour glass was ill suited to this task. Some of them noticed that their pulse was somewhat constant and always available so they used it. Now, back then, things like hours, minutes, and seconds were not firmly defined. The day was, for some reason (that's another discussion) thought of in terms of 24 hours, but a summer day had longer hours because that was 12 hours from sunrise to sunset. And the summer, night hours were shorter. And winter reversed that with it's shorter days and longer nights Minutes were probably not even thought of early on in this process.
But if the beat of the philosopher's heart was to become one second, then there were around 3600 heart beats in an average hour. Not that anyone thought of it that way at first, but still, that number was about right. So when those philosophers and tinkers started to make clocks, they already had the variable sized hours and the human pulse for the second. Now 3600 is an awful large number to display on a clock face, so it needed to be divided into smaller unit(s). 3600 = 60 X 60: That's real even and bingo, the minute was born. This had several advantages. First, clocks needed gears and you can divide the seconds by 60 twice with the SAME pair of gears. That appeals to anyone who is making the parts. And 60 is also a number with a lot of prime factors: 60 = 2 x 2 x 3 x 5. So it is easy to work with when further subdividing the time interval. You can easily get halves, thirds, fourths, fifths, sixths, tenths, etc. No other relatively small (less than 100) number of divisions has all of the first six integers (1 to 6) as it's factors. So 60 minutes to an hour and 60 seconds to a minute is a reasonable choice.
OK, but we were talking about angles, not time.
The earth is a sphere, well pretty close anyway. And it revolves in 24 hours (above). One revolution and one circle are about the same thing. So we have 24 hours x 60 minutes x 60 seconds in time measure for the Earth's single rotation. And again, those philosophers liked to use numbers that could be divided by a lot of factors so 60 would have a lot of appeal to them. The hexacontade, which had 60 divisions, is a unit used by Eratosthenes. However, 60 divisions is a bit on the coarse side and the guys who were surveying land or hurling stones at castle walls from catapults or steering ships on the ocean would want something finer. Why they did not use 24 x 60 = 1440, I can't say. Perhaps that was still a bit too many. Perhaps some did, I don't know. But 6 x 60 = 360 seems to have caught on. 360 divisions were probably around the level of accuracy that ancient devices could measure and/or use for aiming things. And that added two more simple factors to the list of prime factors: 2 and 3. It's prime factors are 2 x 2 x 2 x 3 x 3 x 5 and it can be evenly divided by: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and of course, 360. They loved it.
Anyway, just like the spacing of our railroad tracks can be traced back to the ancient Roman cart makers, I suspect that 360 degrees in a circle can be traced back to ancient philosophers using their heart beat for a measure of short time intervals.
And also for their love of numbers that can be evenly divided by a lot of factors. This is one drawback of the metric system, which is based on the human hands having ten fingers (thumbs included). But 10 has fewer factors than numbers like 12. So 10 of anything can only be divided by 2 or 5 while 12 can be divided by 2, 3, 4, and 6. Put that in your metric pipe and smoke it. Incidentally the Grad, which is 1/400th of a circle was an attempt at creating a metric unit of angle. It has 100 Grads in a right angle. And no one whom I have ever met has ever used it. There really is some sense to the older, more divisible English units.
Very interesting question! And it begs the question, "Why does metric use 360 degrees?" Why not 100 or 1000?