Accuracy in any rifle depends almost solely on the condition of the barrel. If the barrel is right, its going to shoot. Most of the low priced commercial guns have barrels made in Italy or Spain, often by the same companies. Lyman, and Cabela and Dixie have the guns made at the same plants. The barrels are pretty much alike.
You pay more for a barrel made over in the USA because of wage differences, and the fact that US barrel makers Have to make a better barrel to compete for the foreign competition. Small producers offer barrels that have been lapped and choked at a premium price to pay for the extra labor to do this. If you don't know how to do that yourself, you are way ahead paying them to do it for you. Yes, it does make a more accurate barrel.
A retired barrel maker indicates he uses air gauges to check his barrels, ( and those he buys from other makers now,) but he also has some glass rods about 1 foot long that were made to bore dimensions. You can have a barrel that has perfect air gauge dimensions, but is not straight. The glass rods are used to make sure the bore is straight.
A few barrel makers offer back bored barrels, where the rifle is choked a few thousands of an inch over the entire length of the barrel. This is now fairly common on modern breechloading shotgun barrels for the clay target shooters, and the quality of the patterns fired from these new barrels justifies the extra money paid to get them. In MLs, YOu can hand lap the bores of any barrel you buy, and achieve similar results. As far as straightening a barrel however, that is beyond most of our abilities. Because the barrel is everything to accuracy, spend your money on a good one.
Next to the barrel, good bedding of the barrel to the stock is also important. The traditional way of bedding a barrel is with paper shims glued to the channel in the stock. Today, we use epoxy resins, and both improve the bedding,and strengthen the stock. There are those who prefer to " free float" a barrel just as is done with modern barrels, both for target guns, and now for hunting guns. Others think that have good contact between the barrel and the stock the full length of the stock is what makes for accurate shooting. There are accurate rifles made both ways. Your choice on this issue.
Of course, you need good sights, the tang of the breechplug needs to also be bedded properly, and either the rings that hold the barrel to the stock, or the pins or keys must fit properly to get good accuracy. i saw a new rifle, years ago, that managed to warp the stock over the winter, and began spreading balls all over the target. One of the barrel keys was forcibly removed, and the gun went back to being an absolute tack driver, both off the bench, and off-hand. It was a foreign made barrel, but the problem was that the wood used in the stock was green, and when it dried, BOY did it MOVE! The owner later filled in the key slots in the forestock, and then cut new ones, and moved his escutcheons. You had to have the repair pointed out to you to see it.