Sure, it can (and has) been done, but a Mosin-Nagant really isn't the most "sporterization-friendly" design available (Mausers and their variants seem to be more popular).
Recahmbering would be difficult because the 7.62x54r has a case head diameter of .570 which is much wider than most other common rifle cartridges. The only relatively common cartridges I could find in my loading manuals with case head diameters even remotely close to that are .378 Weatherby Magnum and .416 Weatherby Magnum both of which would be far too long for a M-N action. In addition to the case head issue, the single column magazine of the M-N would require pretty extensive modification of replacement in order to accommodate any other reasonably common cartridge.
You've also got barrel length to contend with. Most sporting rifles have barrels in the 22-26" range which conveniently happens to be about the same length as a Lee-Enfield No. 1 Mk. III or No. 4 Mk. I, M1903 Springfield, P17 Enfield, or Kar98K Mauser. The Mosin-Nagants on the other hand have either longer barrels in the case of the 91/30 or shorter barrels in the case of the M38 and M44. In my opinion, the M38 with its 20" barrel and lack of a bayonet lug would make the best platform for a sporter because it's not that short. The 91/30 barrel is, IMHO, just too long and would look ungainly in a sporter stock while the M44 would look odd unless the bayonet and lug were removed. Unfortunately, the M38 is the most expensive of these three variants.
If you decide not to rechamber, there are other issues which make "sporterizing" a M-N somewhat problematic. The split-bridge Mannlicher-type reciever makes mounting a in the traditional position more difficult as you can't have the traditional rear base and must instead resort to a side mount. Also, the farther forward position of the straight bolt handle means that you have to both turn the bolt handle down and use a fairly high mounted scope. Also, you have to make sure that the scope is mounted high enough for you to still have room to manipulate the safety as I've found you have to have a fairly firm grasp on the cocking piece in order to engage or disengage the safety (a feature that seems to have been and afterthought in the design) You can of course use a forward mounted "scout" setup for a scope, but that doesn't really fit the classic "sporter" profile.
Finally, you have the stock. Because Mosin-Nagants aren't popular rifles to sporterize, your options for aftermarket stocks are somewhat limited. For the stock, you've basically got three options: modify the existing one (lots of work and it still probably won't look like a traditional sporter), use a plastic stock like an ATI (cheapest and easiest route but ugly in many peoples eyes), or have someone custom make a wooden stock (spendy).
Basically, in order to really sporterize a Mosin-Nagant, your probably looking a large enough investment of both money and time that you'd be better off just buying a new Savage that's already set up as a sporter. If, on the other hand, you just want to drop the rifle into a plastic camo stock and stick a scout scope on it, and load it full of Wolf JSP's to deer hunt with, well then you'll have a pretty decent deer gun for not a lot of money (though I really don't see anything wrong with a M-N as a deer gun in its original form). Likewise, if you just want to be the only guy at the range with a fancy Mosin, its your money so spend it in whatever way makes you happy.
Recahmbering would be difficult because the 7.62x54r has a case head diameter of .570 which is much wider than most other common rifle cartridges. The only relatively common cartridges I could find in my loading manuals with case head diameters even remotely close to that are .378 Weatherby Magnum and .416 Weatherby Magnum both of which would be far too long for a M-N action. In addition to the case head issue, the single column magazine of the M-N would require pretty extensive modification of replacement in order to accommodate any other reasonably common cartridge.
You've also got barrel length to contend with. Most sporting rifles have barrels in the 22-26" range which conveniently happens to be about the same length as a Lee-Enfield No. 1 Mk. III or No. 4 Mk. I, M1903 Springfield, P17 Enfield, or Kar98K Mauser. The Mosin-Nagants on the other hand have either longer barrels in the case of the 91/30 or shorter barrels in the case of the M38 and M44. In my opinion, the M38 with its 20" barrel and lack of a bayonet lug would make the best platform for a sporter because it's not that short. The 91/30 barrel is, IMHO, just too long and would look ungainly in a sporter stock while the M44 would look odd unless the bayonet and lug were removed. Unfortunately, the M38 is the most expensive of these three variants.
If you decide not to rechamber, there are other issues which make "sporterizing" a M-N somewhat problematic. The split-bridge Mannlicher-type reciever makes mounting a in the traditional position more difficult as you can't have the traditional rear base and must instead resort to a side mount. Also, the farther forward position of the straight bolt handle means that you have to both turn the bolt handle down and use a fairly high mounted scope. Also, you have to make sure that the scope is mounted high enough for you to still have room to manipulate the safety as I've found you have to have a fairly firm grasp on the cocking piece in order to engage or disengage the safety (a feature that seems to have been and afterthought in the design) You can of course use a forward mounted "scout" setup for a scope, but that doesn't really fit the classic "sporter" profile.
Finally, you have the stock. Because Mosin-Nagants aren't popular rifles to sporterize, your options for aftermarket stocks are somewhat limited. For the stock, you've basically got three options: modify the existing one (lots of work and it still probably won't look like a traditional sporter), use a plastic stock like an ATI (cheapest and easiest route but ugly in many peoples eyes), or have someone custom make a wooden stock (spendy).
Basically, in order to really sporterize a Mosin-Nagant, your probably looking a large enough investment of both money and time that you'd be better off just buying a new Savage that's already set up as a sporter. If, on the other hand, you just want to drop the rifle into a plastic camo stock and stick a scout scope on it, and load it full of Wolf JSP's to deer hunt with, well then you'll have a pretty decent deer gun for not a lot of money (though I really don't see anything wrong with a M-N as a deer gun in its original form). Likewise, if you just want to be the only guy at the range with a fancy Mosin, its your money so spend it in whatever way makes you happy.
