I think wedlock is close. However, it is likely less than 276.8 because:
1) I assume you can't have fractions of a ping-pong ball and
2) If you superimpose a 96 inch circumference sphere on a pile of most densely stacked [face centered cubic packing] ping-pong balls of circumference 4.9634 inches, some of the ping-pong balls are going to be cut through by the surface of the sphere.
Example: The diameter of the 96 inch sphere is about 30.5577, and the diameter of the ping pong ball is about 1.58, so about 19.34 ping pong balls fit along the axis of the big sphere. Except you can't have a fraction of a ping pong ball, so you can only have 19 balls.
I don't know how to figure out how many fractions of ping pong balls get cut off. But since 19/19.34 = .9824, I'm guessing that since 276.8 x .9824 = 271 and change, the correct answer is closer to 271.