I believe the set of points (x,y) such that $x y = ±1$ satisfies the conditions.
The set consists of 4 segments of hyperbolas. Any straight line crosses at least 2 of these segments resulting in 2 to 4 intersections. Except for the lines x=0 or y=0 which cross none.
Here's a graph:
Okay.... I googled "A game that doesn't exist" and wound up with "Polybius (square)" which also came up in the commentary below. Hadn't heard of that before.
A game that has existed for over 1500 years" presumably refers to Chess. I thought that might be the 'key' for a simple version of the square or an 8x8 square but it didn't work,...
It is not possible with any finite or countably infinite set.
Take any point P in the set. For each other point Q in the set, draw a line between P and Q. Then take the angle between line PQ and the horizontal axis. This forms a set of angles A.
If S is countable, then the set of angles A must also be countable (since there is at most once angle per point ...