I'm an amateur long range precision shooter, so maybe I can lend a little theorycraft backed by some observed factors in the realm of high precision rifles. I'll use .308 as my default caliber because it's a very popular long range round and the most common military long range round (aside from the .338 Lapua used by extreme range shooting - but I have much less experience and data on hand for that). There are a couple main factors that affect a barrel's precision: length, rifling, and the incredibly important chamber. I'll break down the basics of how each affects the consistency and accuracy of each shot.
Length: This has two major factors included, being the velocity of the round and the stability. A longer barrel means tiny movements by the shooter using a support will affect muzzle direction in smaller increments. Less movement, more consistent and accurate shots, though this is a minimal factor. The velocity comes from allowing the gas to expand in a confined area for longer and get the maximum acceleration on your round. In a .308 you reach the maximum benefit per inch at around 22", gaining only ~17 fps from going to a 24" if you've got a 1:10 twist. Optimal velocity gives you a flat trajectory and optimal spin on each round. This is definitely important, having the right length for your rifle's calibre and rifling will affect accuracy in a huge way.
Rifling: Quality is important here, but realistically with even today's standards in machining you're not going to see too much of a difference between mid range and high end barrels. It's very easy to do this well, and the extra effort that goes into having someone else break it in isn't really necessary. After about fifty rounds, even cheap precision guns like the savage 10 or Tikka t3 lite are really, really great. The most important aspect is the twist rate. Certain bullet weights and velocities like different twist rates to stabilize properly. A .308 using 168 grain match bullets out of a 22" barrel loves a 1:10 twist. Any less and you lose accuracy past about 500 meters, any more and you lose accuracy at all ranges as it rotates too quickly and destabilizes itself in the air.
Chamber: This one is huge. Having proper spacing means the bullet seats itself perfectly every time, or the round doesn't feed. Removing any yaw from the bullet before firing means you are getting a perfectly consistent shot every time you pull the trigger, and consistency is accuracy. It limits your cartridge options, I know some 7.62 and .308 rounds that don't fit into my favourite budget gun the savage 10, but finding the right one can bring you to .3 MOA at 300 meters. The rifle is more accurate than I am, and that's great.
Hopefully this helps explain why a custom barrel, especially with handloads custom made for your rifle, would justify in game benefits. There are a lot more factors, but they're mostly covered by the rules (custom grip, electric triggers, so on).
Cheers