I've thought this through and I think the Gerald Bull supergun HARP/Project Babylon approach is more cost-effective.

8000km/hour is 2200m/s. Reaching 2200m/s at only 10000 gees takes 23 milliseconds, which is only 25 meters if you're traveling in a straight line. So you might be able to do this with, say, a 2-meter-diameter by 25-meter-long supergun, with a total volume of 79 cubic meters, which you might want to dig into the ground to reduce the risk of explosion. Accelerating a 200 kg launch vehicle at 10000 gees takes 20 meganewtons; over a 2-meter-diameter area that's about 6 megapascals of gas pressure, which is only 900 psi, eminently achievable with a light gas like hydrogen (the SHARP gun reached 3km/s). Then it's just a matter of timing the gas release through different ports as the launch vehicle moves through the barrel, a problem that's enormously easier now than in the 01960s with HARP or in 01918 when the Germans solved it for the Paris Gun.

The vehicle itself doesn't have to be 2 meters in diameter, and shouldn't be; an APDS-like approach with a lightweight sabot that fragments upon exit from the barrel gives your launch vehicle better power to penetrate the atmosphere.

Contrast that with a centrifugal launcher like the SpinLaunch design. For the centripetal acceleration to be only 10000 gees at 2200 m/s, the radius of the spinning arm needs to be 49 meters, which means its diameter is 98 meters, four times the length of the supergun. It's a building the size of a city block! But turned up on its side, vertically, so it's 40 stories tall. Its total cross-sectional area is 7700 square meters. You probably have to evacuate the interior so air resistance doesn't melt your rotor, since its outer edge is spinning at Mach 6.7. (This is the "300-foot diameter steel vacuum chamber" mentioned in the article). Even if you can make it only 100 mm thick over most of its area, it's 770 cubic meters, about an order of magnitude more volume than the supergun, and that entire huge area has to be leakproof, which makes it expensive (though admittedly it only has to withstand one atmosphere of pressure, not 60 like the supergun barrel).

From appearances their test launch facility is only about 7 stories tall, so only about 20 m, and they say they're only doing launches at about 450 m/s, which would work out to about 2000 gees by the same math. But it also looks like it's about 3 m thick, 30 times more volume (per unit area) than I described above: 20000 cubic meters at full scale.

The usual problem with centrifugal weapon systems doesn't apply here, though: you don't need to aim your launch vehicle precisely at a target to milliradian precision, you just need to hit the exit port instead of the solid launch-chamber wall. (The enormously larger radius also means you're spinning at many fewer radians per second.) And the free breaking length of carbon fiber is 400 km at one gee, and thus 40 m at 10,000 gees, and the outer part of the rotor can be thinner than the inner part, which is also under proportionally lower acceleration, and there are another half dozen materials with free breaking lengths over 150 km, so a 49 m rotor that spins at this speed is straightforwardly achievable. So I think the SpinLaunch approach will work if they spend enough money on it. I'm not a SneerClubber, I'm not here to sneer.

(Realistically the most likely way for centrifugal space launch to happen is that NASA will cut their funding before they can do their first launch, and ten years later China will build a working centrifugal launcher, because building new things is not really a thing people do in the US anymore.)

But if you're willing to evacuate 20000 cubic meters to launch your vehicle at 2200 m/s, you could drill your supergun barrel down through 6 km of rock, dropping the necessary acceleration to only 42 gees. That would enormously simplify the design of the launch vehicle, though it still wouldn't allow crewed spaceflight.