Caribou is right about some of the changes needed and even more right that it works as it is quite well.
Financially, one of the better things you can do with cheap $200 Mosins in Utah is to buy them, hold them for about until 2020 election time, and then sell them. Bet you could probably get a 50% return on them in return for holding them a few months.
As always, this is a free country and people are free to do what they will with their property absent causing others direct harm. One more chopped up milsurp makes mine more valuable and so on. YMMV and so forth.
Now, IMHO,
This is not really directed at the O/P as much as other people jumping on the idea via an internet search and thinking it would be a swell thing to chop up a rifle just like Granddaddy's Mauser sporter. On the THR, you see such pop up when they inherit granddad's military rifle or buy one cheap and think what a cool thing to do would be chop it up with little idea of suitability, costs, and function. Some do a little internet research, get a THR link to the conversation and post their ideas here. The THR always welcomes new folks and forum posters are more than willing to share knowledge. In many cases, the additional knowledge leads to a change of thinking about how easy and great such a conversion would be. In almost all cases of unmonkeyed with rifles, it is cheaper to sell the rifle, take the money, and buy what they really want.
Those that do not seek guidance initially, usually jump in with more enthusiasm than knowledge and butcher a fine rifle and often just abandon the project after they find it more than they can tackle or spend. I know, because I used to restore them to full military trim after buying these aborted sporter projects that were not too far gone. I quit recently other than completing a few remaining projects due to increasing costs to obtain parts to restore them and medical issues. Even I passed on receivers bored so full of holes for scope mounts that it was more holes than metal, barrels cut off with a pipe cutter, or where people ground on Mauser feed ramps to get space enough to feed the latest hot cartridge too long for the magazine, and so on. Enthusiasm is no substitute for concrete knowledge of firearms, machining and metallurgy, and mechanics especially when dealing with a dangerous product.
To illustrate, one of the problems with the O/P's idea is that such a conversion may never feed right unless you are a machinist willing to tinker and make your own parts. The magazine is designed for a rimmed cartridge and has certain features to prevent rimlock and jams with an interruptor being a key feature and the 7.62 Russian has a different case shape than the Mauser derived .308. Feeding will be an issue unless you want a single shot.
Bannerman did an ill-advised conversion of Mosins in the 30's to .30-06 that was simply a blowup waiting to happen. The 8mm Mauser trick was done as an expediency during interwar years and sent to countries that counted their soldiers lives cheap. Neither are really that functional either. One could chamber it in 8x56r but you would need a new barrel as there probably is not enough meat in the Mosin left to go from .311 to .323 or so and the ammo sold and its function is inferior to that of the original cartridge--so what would be the point.
I have come across somewhere where someone converted a Mosin to fire .303 British or the 8x50r or other obscure cartridges but then again why?
The 7.62x54r, as it is, already is of the more versatile cartridges to handload for on par with the .30-06 and over here, the bullet selection for handloading is generally better than that of the 8x57 Mauser. They also shoot well with cast.
Some info on bad conversions.
http://mosinnagant.net/global mosin nagants/8mm_blindee_converted_mosin_naga.asp
Enjoy the Mosins as they are--you can buy match ammo for them at cheap enough prices, they can be handloaded to perform multiple roles just as the old .30-06 can be, and if a person wants a bit more accuracy (although Mosins can be pretty accurate) then get a Finn Mosin with a .308.
Mosins are not Mausers where a simply barrel change can allow a whole family of cartridges to be used. Rimlock and feeding problems for Mosins with their issued ammo can be an issue which is why the Finns made some changes to keep the rifles from jamming.
They are not an Enfield which is also somewhat difficult to adapt to rimless cartridges but worked ok for the Brits later with their No. 4's to 7.62 Nato after upgrading their metallurgy and I have heard of some souls converting them to .223.
If people get a Mosin for $200 to sporterize, you will have a Mosin sporter that costs a bunch in time and worry to get it to function half-butted, spend a fair amount on a new stock and then inlet it properly, have to D&T with scope mounts, get a new barrel and fuss with the magazine feeding, alter the bolt, extractor, etc. so you get some reliability and function, worry about safety if you fire commercial cartridges as the old Mosin was designed to handle a rimmed cartridge to seal the chamber from gas events and otherwise does not have good venting for a rimless cartridge that it was never designed for.
At the end, you will have a rifle that is inferior to modern rifles in function and accuracy, an oddball sporter that probably will never get what you paid for out of it as the collectibility is gone, and the conversion might possibly compromise your safety. In addition, you see the shame when the original Mosins left untinkered with go up in price, even the cruddy ones, greater than what you paid for your pristine new rig.
Want to sporterize one, then skip the full trim military rifles and look for a barrelled action or a butchered half-butted sporter.
I've seen enough bad Mauser and Enfield conversions (one person converted a No. 1 rifle to .300 H&H, I saw another milsurp converted to a .257 Weatherby, and so on.) to last a lifetime in the gunsmithing house of horrors.
Want a .308--CDNN is selling Thompson Center Compass rifles for $269 with probably a better trigger, much easier scope mounting, efinitely better metallurgy, a warranty, and most probably considerably more accuracy than the old Mosin platform could give outside of expensive MTU type optimization. Or skip that and get an Axis, Axis II, Remington starter, Thompson Compass, or if you are feeling rich, a Ruger American for such that would probably best all of them.
Want a milsurp .308, buy a large ringed Mauser receiver for about $100, get a decent blaster grade barrel from Brownells or Midway, put a mag block in it and Mauser sporter stocks, including rather nice ones, are cheap. You'll get a tough, reliable sporter, that will probably, in the end, shoot more accurately, feed better, and require minimal alterations to get it to work for less time, money, and headache than getting a Mosin to work in .308. Plenty of attractive sporter parts for Mausers are around and the Mauser is a simpler system to adapt to feeding new cartridges, especially cartridges derived from original Mauser ones.
The golden age of milsurps was because a lot of people had the machining and woodworking experience with not much money and milsurps were far cheaper than the commercial rifles and about on par with them for accuracy. They were also willing, in some cases, to roll the dice with uncertainty on safety due to general lack of knowledge of pressures, etc. Now, milsurps are scarce and getting scarcer, their ammo for the less popular rifles is difficult to acquire or dangerous to shoot, and commercial rifles are cheaper than them with superior technology and made for modern high pressure cartridges.
The new golden age, if you want to tinker with firearms, and do so in a safe way is to go to the dreaded AR platform as it is truly the Lego version of firearms and there is always the AR-10 challenge for those wanting bigger cartridges. The metallurgy is superb, ammo is generally available along with brass, powder, and bullets, for reloading, parts are plentiful and cheap, etc. And, unlike those old warhorses, it is easy to convert to a new cartridge and sell your old upper to someone else while keeping your perfect trigger and stock, etc. You will have accuracy, cheap repair parts, excellent barrels, huge assortment of magazines, excellent ergonomic stock designs, and so on. The AR system leaves the old idea of a switch barrel Mauser in the dust as far as ease of cartridge conversion--how much easier is it to simply put an new upper on an existing lower? Or to switch out a bolt head? Buffer weight? Magazine capacity? Scopes? Stocks?
Want to save history, then honor the rifles in the condition that they are in, fix those in poor shape, and accept their limitations. Keep Grandpappy's rifle as it was and respect the memory of him or if buying one, respect that many folks wielding that rifle, had to sacrifice their time, perhaps injury, or even their life in wielding it. Absent maltreatment or the machinations of evil gun grabbers, that concrete example of the past will persist even after every poster on the THR is gone to their reward.
I hate to quote the bad guy in Indiana Jones, but he is right in that we are simply passing through history, the examples of our struggles left behind in artifacts and memories ARE History.