spasticteapot

If I were you, I'd use a Blade paintball gun (est. 20$) instead. I recall seeing a version of one modded for Nerf that could fire a modified (weighted) dart up to 150' accurately. They've got a decent regulator, and can get forty shots of of a fifty cent CO2 cartridge.
Better-fitting darts might also help.

Ah, so that's what first is doing this year.

8" balls are more like dodgeballs than nerf balls.

The ball can't move faster than 26 mph? That's like...38 FPS. Nothing.

You could try some kind of pneumatic ram driven cannon, with one big barrel that has a magazine coming off of it, and a rod (should be light) inside it that's driven by a pneumatic ram. When it's retracted, the mag is open, and a ball feeds into the barrel. When the "gun" fires, the rod pushes forward and shoots out the ball. I have not personally built a gun like this, so I don't know the functionality of it. It might be liable to jam with soft projectiles.

It's a shame the ammo is so big. If you were shooting the 2" nerf balls or something, you could just have 10 small pneumatic spudguns. Since the speed is so limited, they could be really small.

On a side note, what makes the gun you linked to "semi-auto". It's bolt action.

I was thinking something similar, but mechanical. Of course, everyone else is hung up on spinning wheels, with no way to reload.
(we're limited to Bimba pneumatics, sadly; and a high-speed long-throw cylinder would cost a mint.)
That said, a PVC cylinder akin to those you have built might do the trick. One would only need about 10 PSI to fire, too, assuming a 1" diameter rod. (31.4 lbs. of force should do the trick!)
Got any links to anything like this?