The Elements opens with twenty-three definitions, and the first is four words in Greek and seven in most English translations: a point is that which has no part. Nothing is added. Euclid does not say what a point is made of, where it is, or how many there are; he says only that it cannot be divided.
That is a stranger claim than it looks. Almost every other object in Book I is built, and Euclid tells you how to build it. The point is not built. It is assumed, placed, or found where two other things cross. This makes it the only primitive in the whole system that the two classical tools cannot produce on their own, because both tools consume points rather than making them: a compass needs a centre and a radius, and a straightedge needs two points to join. Constructions begin with points already given.
Where new points do come from is intersection. A circle crossing a circle, a line crossing a circle, a line crossing a line: these are the three events that generate every fresh point in a classical construction, and the algebra of exactly those three cases is what later proved certain figures impossible to draw. The whole theory of constructibility is, at bottom, a theory of which points can be reached from the ones you started with.
A point has no length, no breadth and no interior, so it has no size at all. It has position and nothing else. Mathematicians after the nineteenth century largely abandoned the attempt to define it, and treated point, line and plane as undefined terms fixed only by the axioms relating them. That is the modern position: a point is whatever behaves like one.
None of this is evasion. It is the recognition that the starting element of a system cannot be explained in terms of the system it starts.