Accuracy...
Dad of 6--You said,
What does accurate mean to you?
And I have to respond, "it depends." Now, for a target-competition rifle, accuracy needs to be of the one-hole-at-100-yd variety, and that hole had better be rather small, and we're talking 10-shot groups here. A rimfire competition rifle will open up a bit @ 100 yd, but it had better not open up much if it wants to stay in my arsenal. For benchrest competition, a rifle that can't hold 0.25 MOA or so is useless. For a hunting rifle, 1.5 MOA is acceptable, although I'd prefer tighter than that. Same goes for a plinking rifle. For a handgun, it hardly matters at my skill level, but let's say for the sake of argument, a handgun needs to be capable of putting 10 shots in a row, into the x-ring, in slow fire. (Understand,
I am not that capable, but it's me that is the limiting factor, not the handgun.)
With a shotgun, again, my skill is the limiting factor, and IMHO, the fit of the shotgun to the shooter is more important than the shotgun's "accuracy" in hitting game or breaking clay birds. With slugs, a shotgun just becomes another kind of hunting rifle, and the rules for those apply.
Any of the above might be that accurate, out of the box, or might be made that accurate, with tweaking by the owner. I don't buy many brand-new firearms, so most of mine get tweaked, either before I bought 'em or after.