Assuming you are talking about modern rifles with modern designs and reasonably well made parts, that is anything made in Europe, North America, and Japan, then:
* Action plays a relatively minor role. All modern rifles have good enough action. High end single short actions still makes a difference but that difference will not be noticeable before you have a good barrel. People often put a custom barrel on a factory action to get good results. For example, Shilien makes barrels that can be installed into a Savage action by any average shooter, and it is a popular upgrade for many savage owners in the States (oddly, no one seems to do it here). alsmo many people put Lilja barrels on CZ 455 actions. The contrary is not true. No one puts a factory barrel on a custom action. I have never seen a Ruger factory barrel on a Volquarsen or KIDD action, for example. quite simply it would have sucked just the same.
* Barrel plays a major role. Not all modern rifle makers make equally accurate barrels. Quality of blanks, method of making of the rifling, precision of fitting , lapping the barrel all makes a difference -
- You will see that some guns makes and custom barrel makers advertise the material of their barrels, or that they use well known branded blanks.
- super custom barrels make rifling by cut rifling, it is slow and expensive, but you can make rifling of any twist rate and number of grooves. Rifling of most mass produced modern barrels are hammer forged. It is relatively less precise and less accurate method. It also introduces stress to the metal. Tikka is the only one that makes consistently accurate hammer forged barrels. Button Rifling is the middle ground. It results in more accurate barrels than hammer forge. Custom barrel makers tend to use button rifling, for example Lilja, Trueflight, and Shilen. Savage is the only mass production maker that uses button rifling.
- In terms of barrel fitting, thread fit is generally better than press fit. Press fit can be very good too - if super tight tolerance. Anschutz uses press fit and wins more Olympic gold medal than everyone else combined. Cheap Remingtons also press fit their barrels and they are apparently pretty bad.
- Lapping barrels makes the bore smooth and shiny, the bullet travel through the barrel uniformly and consistently, improving accuracy. But to do this right it has to be done by hand and it is time consuming and therefore costly. This is similar to high end knife makers always sharpen by hand while mass produced knives just get grinded. I think CZ is the only factory rifle maker that hand laps their barrels, which is probably why it has enjoyed a reputation for accuracy. Lilja hand laps every barral. True flight only hand lap their highest grade barrels.
* Bedding is a controversial topic. I personally think that bedding is similar to action in terms of how much it affects accuracy. Most modern rifles has good enough bedding. Most Tikka with plastic stock and no bedding can do 1 MOA easy. Some modern rifles use alternative method to achieve better stock / action surface rigidity. For example, Savage has been doing this aluminium action/stock fitting thing that they call accu-stock and it is fitted in most of their rifles with cheap plastic stock.
* Bullet makes a big difference, especially between the bulk ammo or mil surplus ammo and proper high end ammo or hand loads. An Olympic grade single shot shooting Winchester 555 pack will be out-shot by a CZ452 shooting Eley Ten-x.
* Stock to me ranks below bedding and action. I feel most people upgrade stocks for better customization, or lighter weight, or better carrying ergonomics, or aesthetics rather than inherent accuracy improvement.
* Trigger ranks above barrel in accuracy impact for free hand shooting. I do free hand shooting 95% of the time. Personally I shoot far better with a super light trigger and average barrel than i would with a super accurate barrel and average heavy trigger. But if you shoot off a bipod, then trigger makes less of a difference once it is below a certain threshold. For example, I think going from 7 pound to 3 pound would make a much bigger difference than from 3 pound to 1 pound.
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