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Nah, bullets don't go anywhere near escape velocity. Escape velocity is ~11.2km/s and the fastest bullets (FAR faster than most) only go ~4000f/s, which is barely over~1.2km/s.
Any bullet that is shot up will come back down, and not terribly far away, either. Even the biggest artillery systems only have barely over 100km range.
What if my gun is a multi kilometre long railgun?
You're compensating for something.
The US government tried that a long while back... The company Spinlaunch is currently working on yeeting stuff into orbit with a centrifuge... So yes, some unusual methods can work.
The moon isn't at escape velocity either (source: It's still there).
Doesn't really change the numbers probably, but you'd need a little less than 11.2 km/s to reach the moon.
Wrong, that is not how orbital mechanics work. The moon IS below escape velocity, but it's orthogonal to the force of gravity. It also has a 240000 mile head start on getting away, yet it's STILL not escaping while traveling over 1km/s.
Shooting a bullet straight up, you would have to shoot faster than escape velocity for it to even reach the moon when using simple ballistic calculations.
There is A LOT of energy in those thousands upon thousands of miles.