The engineering required for an orbital launch gun is beyond the scale of anything yet built.
Let's say, for the sake of argument, that your space gun's acceleration track climbs the slopes of Chimborazo in Ecuador and releases the launch vehicle at the summit.
From there, if you launch eastward, you get about 460 m/s for free from the rotation of the Earth near the equator. LEO requires about 7800 m/s. Earth escape requires about 11180 m/s from that altitude.
So if you go to LEO, you need another 7340 m/s of velocity. If you accelerate at 3 g, your track needs to be 920 km long. That isn't quite the distance from the end of your track to the Galapagos Islands.
The same length of track could hit escape velocity with constant acceleration at 6.5 g.
For reference, the Large Hadron Collider is 27 km in circumference, and cost about $5 billion. A space gun would be 34 times longer, and portions of it would have to be evacuated and neutrally buoyant at some depth under the ocean. The entire length would need to be powered.
If you built a circular track 920 km in circumference, 147 km radius, the centripetal acceleration required to keep something moving at 7340 m/s on that track would be more than 37 g. That only increases as you reduce the radius. So no getting away with a shorter track.