The short answer is that it is a plausible mechanism for g-hardened payloads, but probably not for soft payloads (like human beings). The problem is that accelerating up to orbital velocity at the maximum survivable continuous rate (~4 or 5 g) still takes hundreds of miles, which is probably an infeasible length for a barrel. Additionally, if it's not somehow supported so that it rises above the atmosphere, the de-acceleration from drag at the exist of the barrel is also too extreme.
https://www.youtube.com/watch?v=Moo5nuLWtHs
The short answer is that it is a plausible mechanism for g-hardened payloads, but probably not for soft payloads (like human beings). The problem is that accelerating up to orbital velocity at the maximum survivable continuous rate (~4 or 5 g) still takes hundreds of miles, which is probably an infeasible length for a barrel. Additionally, if it's not somehow supported so that it rises above the atmosphere, the de-acceleration from drag at the exist of the barrel is also too extreme.