Why is a full circle 360° degrees? What's the reason we agreed to setting the number of degrees of a full circle to 360? Does that make any more sense than 100, 1000 or any other number? Is there any logic involved in that particular number?

As it has been replied here - on Wonder Quest (webarchive link):

The Sumerians watched the Sun, Moon, and the five visible planets
(Mercury, Venus, Mars, Jupiter, and Saturn), primarily for omens. They
did not try to understand the motions physically. They did, however,
notice the circular track of the Sun's annual path across the sky and
knew that it took about 360 days to complete one year's circuit.
Consequently, they divided the circular path into 360 degrees to track
each day's passage of the Sun's whole journey. This probably happened
about 2400 BC.
That's how we got a 360 degree circle. Around 1500 BC, Egyptians
divided the day into 24 hours, though the hours varied with the
seasons originally. Greek astronomers made the hours equal. About 300
to 100 BC, the Babylonians subdivided the hour into base-60 fractions:
60 minutes in an hour and 60 seconds in a minute. The base 60 of their
number system lives on in our time and angle divisions.
An 100-degree circle makes sense for base 10 people like ourselves.
But the base-60 Babylonians came up with 360 degrees and we cling to
their ways-4,400 years later.

Then, there's also this discussion on Math Forum:

In 1936, a tablet was excavated some 200 miles from Babylon. Here one
should make the interjection that the Sumerians were first to make one
of man's greatest inventions, namely, writing; through written
communication, knowledge could be passed from one person to others,
and from one generation to the next and future ones.  They impressed
their cuneiform (wedge-shaped) script on soft clay tablets with a
stylus, and the tablets were then hardened in the sun.  The mentioned
tablet, whose translation was partially published only in 1950, is
devoted to various geometrical figures, and states that the ratio of
the perimeter of a regular hexagon to the circumference of the
circumscribed circle equals a number which in modern notation is given
by $ \frac{57}{60} + \frac{36}{60^2} $
(the Babylonians used the sexagesimal system, i.e., their
base was 60 rather than 10).
The Babylonians knew, of course, that the perimeter of a hexagon is
exactly equal to six times the radius of the circumscribed circle, in
fact that was evidently the reason why they chose to divide the circle
into 360 degrees (and we are still burdened with that figure to this
day).  The tablet, therefore, gives ... $\pi = \frac{25}{8} = 3.125$.

