As always, there is the necessity of establishing definitions. To me, a 1/4 moa rifle is a rifle where the shooter can say "watch this" and have a reasonable expectation of shooting a 1/4 moa group. I've had some short range BR rifles which were legitimate sub-1/4 moa rifles with which I could do just that and these rifles all shot sub-1/4 aggregates in competition. The thing is, not only is it difficult to make that step from sub-half to sub-quarter; it is so easy for that sub -quarter to turn into a half-plus. Sometimes, all it takes is to enter into a competition with that rifle! Sometimes, things just go wrong.
I remember one time, I was practising with one of my "good" rifles. This was a 6x47 (222 Mag necked up), Hart barrel, on a 40x action, Lee Six stock and was a legitimate sub-quarter rifle. On this day (in 1977) I shot five five-shot groups. Four groups were .2 or less but the third group measured .370. The second shot was a flier and, when the first two shots afe .370, nothing you do will make the group smaller. The point is, although this rifle was certainly a sub-quarter rifle, sometimes it wasn't. So, we just have to establish within what parameters we expect a rifle to perform before we are comfortable calling it a 1/4 minute rifle. For me, the "watch this" scenario is my expectation. If we loosen up the requirements to include the one group fired on a windy Sunday last March, we can have a whole bunch of quarter inchers. I have fired numerous 1/4 inch three-shot groups from my 308 Norma Mag Model 70 but the "watch this" performance level is more like 3/4. If I make it five shots, the "watch this" expectation is more like an inch but I digress. The question was what is necessary to get to that 1/4 level and my answer is, first and foremost, a rifle which has that capability.
As far as headspacing is concerned, I think it is semi-important and I think concentricity is even more so. I have had fire forming loads in 6mm PPC which shot well enough to be competitive. Of course, fire-forming the PPC is done with a crush fit on the parent case so headspacing is good and alignment is good. On the other hand, I've also had good results with new brass in 6 BR's and 308's. In these, head clearance was probably on the order of two to three thou with new brass.
In the end, it mostly comes down to the rifle. I had a friend (Bob Forslund) who told me "If a rifle in a decent caliber shooting a decent load doesn't shoot under 3/8, there is something wrong with the effing rifle (Bob tended toward colorful language)". Nothing I have seen in the intervening forty-some years would lead me to disagree with this statement. Elaborating, Bob said (more or less), "when you see a guy shoot an effing half inch group and he decides he needs to weigh all of his effing brass, then you know you are looking at a guy shooting with his effing head up his effing ass. He needs to fix the effing rifle first". I won't even try to achieve half minute with one of my hunting rifles simply because (a) it is unlikely I can shoot then that well and (b) it doesn't matter. So, if you want to shoot 1/4 moa, you have to have a rifle which can do it.
Someone always brings up the difference between accuracy and precision and I have always looked upon this as so much BS. The truth is, the only difference between shooting a five shot group and shooting five shots for score lies in the first shot. To hit five x's, one has to clip the x with the first shot then continue to do so. To shoot a "zero" group one only has to hit a pre-determined spot (the first shot) four times. If, as sometimes happens, the first shot is a flyer, it is common to "chase" that shot with the subsequent four. If shooting for score, if the first shot is a flyer, the shooter has to forget that shot, unless he is certain there was some atmospheric condition which caused it, and continue to shoot for the "x". If one is holding the rifle, whether prone, offhand or whatever, there is another, huge, factor involved. WH