Why can't humans throw objects at bullet speed when the force involved seems so small? The force that propels a bullet should at least be similar in intensity to the force experienced as recoil and the force exerted on the target. But these forces seem really low (the equivalent of a shove).
If all it takes to make a small object achieve bullet speed is the equivalent of a shove, why can't humans throw objects at these speeds?
 A: As The Photon wrote in a comment, the issue is that you have to accelerate your hand as well.  So while the 4.4g bullet only needs 304N to accelerate (by your numbers in the comments), your 400g hand has to reach the same speeds on roughly the same distances, requiring on the order of 30000N, well outside the range of the human body.
You may, however, be interested in some tools we've developed to improve our mechanical advantage, letting us accelerate things without accelerating our hands at the same time.  The sport of Jai Alai, for instance, has a hand held device called a cesta which lets us more efficiently transfer energy into a ball.  The record for the 125-140g ball was 302km/h (188mph).  This was the fastest human-driven ball until Jason Zuback drove a ball at 328km/h -- a golfball hit by a golf club.
A: The problem is simply that we can't move our arms quickly enough. We have the strength- a lever-action airgun shows that the equivalent of a good shove is enough to fire a pellet- but we don't have the speed.
A: If you could accelerate the gun to the same speed it has right after the shot, you could use it to hit the bullet and make it fly at the right speed. However, this speed is still very large and you cannot use your shoulder hand system to do it. Our body can stop a gun moving at such speed but cannot bring it back to it.
