The regular hexagon has the simplest construction of any polygon, and it requires no measuring whatever. Draw a circle. Without altering the compass setting, place the point anywhere on the circumference and strike an arc across it; move to that crossing and repeat. Six steps bring you exactly back to the start, and joining the six marks gives a regular hexagon.
This works because the side of a regular hexagon inscribed in a circle equals the radius. Each vertex, its neighbour and the centre form an equilateral triangle, and six of those fill the 360 degrees about the centre at 60 degrees each. Euclid gives the construction at Book IV Proposition 15 and notes the same fact.
The hexagon is constructible under Gauss-Wantzel because 6 is 2 times 3: a power of two multiplied by a single Fermat prime. Bisecting its arcs gives the 12-gon, the 24-gon and so on indefinitely. This is also why the hexagon and the equilateral triangle are so closely bound together. Joining alternate vertices of a hexagon gives a triangle, and superimposing both such triangles gives the six-pointed star, whose interior is again a hexagon.
Its interior angle is 120 degrees, the largest of any polygon that tiles the plane alone, and that is the reason three hexagons meet at every vertex of a honeycomb rather than four or six. Among all polygons that tile the plane, the hexagon has the fewest neighbours meeting at each corner, which turns out to matter a great deal for efficiency.