Place 6 darts around the circle's perimeter at equal distances, in the pattern of a regular hexagon.  Place a 7th dart at the center.  Each dart is now exactly one radius away from its nearest neighbor(s).

This is the farthest apart the darts can be from each other since the movement of any of the 7 darts in any legal direction (i.e. still on the dart board) results in that dart getting closer to a neighbor, thus decreasing the smallest inter-neighbor distance.