Research what the mujahideen were doing with the AK47 during the Soviet invasion. There's a reason why it was the gun used in Guns of the South by Harry Turtledove. Also read the Crosstime Engineer series by Leo Frankowski. There were convoluted repeating arms designs in revolutionary war times. Modern guns have all the bugs worked out. Don't expect huge numbers, but with the basic engineering worked out, they're totally doable.

As has been pointed out (repeatedly!), building the guns isn't the problem. Making the ammunition is. It has to be made to fairly tight tolerances, or it doesn't work.

(1) There were no uniform standardized units of measurement. 0.38 inches in this shop was not the same as 0.38 inches in that other shop. (2) The materials were not available. Steel was rare and of variable quality, and you could not go and order it in quantity. They didn't even have a notion of how to measure toughness or hardness or tensile strength. (3) There was absolutely no way to make the primers and propellants used in WW1 ammunition. There was no way to explain what those substances were. (4) A jeweler might have been able to make a working replica of a WW1 rifle. In a year.

I do think that all these "could my primitive people build technology X" questions display a certain deficit of understanding of the complexity of a modern economy. The point above about standardised units is a good case in point. That didn't just happen by accident!

@DrMcCleod just that Beretta did manage to mass manufacture arquebuses in 1550 - because Beretta was one arsenal churning out all the barrels in one manufactory plant. Why did it work? They had one measurement to measure against, and the bullets of arquebuses have about 1 to 2 millimeter of tolerance built in!

It's not directly analogous, but look into Khyber Pass copies of modern firearms: en.wikipedia.org/wiki/Khyber_Pass_copy

@AlexP That is why many early weapons were manufactured using the Gauge unit. While sizes where not very standardized, weight was. Accurate weighing is extremely important in a gold/silver standard based economy; so, gunsmiths used the gauge system which uses fractions of a pound of spherical lead. While my inch might be very different from your inch, we would both own scale weights that were made to very exact standards; so, if I use my wights to make a 12 gauge lead ball and you use yours to make a 12 gauge lead ball, we will both get almost identical balls.

This could still lead to a couple of mm of variance, but not nearly as much as using length units.

POssible duplicate of multiple questions. Could a percussion rifle be produced in the 15th-17th century? and [ how-advanced-could-someone-make-a-firearm-with-medieval-tech‌​nology] (worldbuilding.stackexchange.com/questions/103651/…)

@Nosajimiki: Do you actually know anybody who made pre-modern weapons with calibres precise to one hundredth of an inch? Or anybody who made musket or cannon balls precise to one hundredth of an inch? Pre-modern weapons threw sub-calibre projectiles; calibres were approximations, as were projectile weights; and nobody wasted time and effort to make the musket or cannon balls precisely spherical, for the simple reason that it would have been useless. (Ah, and how on earth do you inted to maintain the sphericity of that soft and heavy lead ball?)

The answer to this question, despite some imaginative responses to the duplicate references, is always "no." Technology is a pyramid, and doing something at the top of that pyramid (whenever that might be referenced) takes pretty much the entire pyramid. Trying to push the top back means pushing the entire pyramid down. The interdependence of technology is massive and astounding. Remember, you need the advances in the sciences and all the manufacturing capabilities. You wouldn't be a pre-industrial society anymore.

@AlexP Seems I was a bit mistaken in terms of calling this an "early" method since it was only standardized in the 1800s, but it was simple enough of a method that that it could be easily introduced earlier in an alternate timeline.

As for how accurate gauging was, medieval and later currency scales were generaly required to be accurate within a carat (~0.0004 lbs) and all scales were measured by central government authorities; so, between 2 nations, they could be different, but each kingdom was generally standardised pretty well. This degree of accuracy along with the accuracy of reaming tools available at the time would make 1/100th of an inch very doable.

... but doable and done are not necessarily the same thing since many barrels were intentionally overbored by up to 0.07" because it could cause a favorable spread in shotgunds

@JBH not necessarily. Chemistry and Industrialization may have happened at about the same time, and they certainly make eachother easier, but neither really relies very heavily on the other. Most chemistry can be performed on the small scale using hand crafted tools, and most industrialization can be done without chemistry through over-engineering with lower quality materials.

@Nosajimiki making weights accurate is easy making machining accurate is very very hard, and that is what modern firearms need, all the parts and bullets in a machine gun need to be made to the same standard or it is more dangerous the the wielder than the target. they also need good chemistry to produce metals of high enough quality, not to mention the powder and primers themselves. you are also very wrong about accuracy of medieval currency, France was known for having 250,000 different units of weight and measure at the time of the French revolution due to the variance from place to place.

@Nosajimiki modern weapons are not modern weapons solely due to chemistry. They require advances in metallurgy and machining including manufacturing facilities that would obviously (and it really is obvious) be used for every other aspect of industrialization. I've had this argument with a dozen people over the last two years, and I always win. The only way for a pre-industrialized society to manufacture anything post-industrial and remain, fundamentally, pre-industrial, is to magically forbid that tech from being used by or for anything else. As I said... obvious.

@JBH Cottage industry does not mean you have no elements of industrialization, it just means your production is split up into small workshops run by home owners instead of centralized factories. There are many cultural or practical reasons a society may choose to do this even if they have the technology for centralized production. Perhaps the civilization puts a strong emphasis on the family unit which makes leaving home to work taboo, or maybe a plague has made centralized production too dangerous, or perhaps your nation is just too economically broken to invest in centralized factories.

Also, you assume that it has to remain that way to fulfill the OPs requirements. In the late 20th century, many poorer nations like China were manufacturing modern hardware for first world nations using machines that were typically over 100 years outdated following the cottage industry model. They did not have the means to make modern factories (yet), but they did have desperately impoverished workers willing to work for next to nothing; so, they used what they had to make modern technology. They were not as efficient as western factories, but they made up for it with the cheapness of labor

@Nosajimiki Ferrari stopped hand-building engines for a reason. Your cottage-industry example is fallacious.

@JBH You are actually proving my point. The OP wants 1910-1940s quality machine work, and Ferrari was using hand-crafting techniques that could compete with industrial machine work up into the 1980s. Here in 2020, hand crafting absolutely can non compete with computer guided precision, but that is not the question. I think "modern" may be throwing you off. While the term "modern" in the vernacular often means the same as contemporary, the "Modern Era" technically ended in 1945.

@Nosajimiki (shakes head) What is the feasibility of producing (on a wide scale)... It's that "on a wide scale" part that's the problem. You can't produce factory-built items at factory-built precision and factory-built speeds without the factories.

@JBH A Tanegashima took a craftsman entire month to make, yet Japan made 300,000 of them. Even if it took 3 months to handcraft a single AK-47 and its ammo, Japan could have cranked out 100,000 of them with the same investment and effort

@JBH I've added citations and clarity to my answer.