Sounds like quite a hard problem. You can't use anything in the actual target area (the whole board), since
a. Any sensor will get repeatedly stabbed to death.
b. Putting anything other than the usual soft board there will negatively affect the 'thunk-nature' of the board, and detract from the game.
So it's a remote sensing problem. You have to detect the location of several metal pins (the dart needles) within an area, to high accuracy.
The darts may rest at angles, but it's the point of penetration that counts, so you have to sense at (or very close to) the surface of the board.
RF/electromagnetic/radar solutions are complicated by the wide variety of dart body materials, ranging from mostly plastic, to all metallic.
Best I can think of is two or more hi-res cameras looking at the board surface from oblique angles, and some distance away. Then it becomes an image recognition software problem. A fairly hard one, due to angled darts, different body shapes, and darts close to each other.