Project Appleseed

If I wanted to shoot say 800 yards with sub 2 MOA (would love sub MOA) accuracy on a serous budget what rifle(s) would I look at?

I found a Russian Mosin Nagant 1891/30 wHex Receiver:
http://www.aimsurplus.com/acatalog/Russian_Hex_Receiver_Model_91_30_7.62x54R_Mosin_Nagant_Package.html

But from my understand those are 4 MOA rifles... (the Finnish are 2-3 MOA but not the Russian).

So I looking for a rifle that:

Sub 2 MOA
Effective to 800+ yards
Cheap to attach Scope
Budget

Typically, you'll not have a service rifle shoot sub 2 Minutes at 800 yards.  CAN you shoot that well?  I know I sure can't.  My 91/30 and I have typically done 10-12" patterns at 200 yards, with surplus light ball ammo.  HOWEVER, they are a wonderful rifle, especially considering the current low prices, of both rifle and surplus ammunition.  An optic can be attached to a 91/30, but irons are a good way to get started.  The 7.62x54R round is certainly effective at 800 yards, but remember 800 IS A LONG WAY OUT THERE.

The important thing is to just get out there and practice marksmanship!  I love 4-6 MOA milsurp rifles.  They are my favorites to shoot.

I think the best you could do on the cheap would be to get something like a Mosin and then spend some time on it to make it more accurate. Possibly the easiest thing to increase accuracy would be loading your own ammunition. For other things to do I would visit a surplus rifle site that specializes in that rifle and see what advice they could give.

I own(ed) around a dozen mosins of various manufacture and condition and have found that they shoot anywhere from 1-6 moa's.  My one moa rifle was a Finnish.  I got rid of it because it had a bulge in the middle of the barrel, but it shot great.  My current Finnish mosin will only do 6 moa with Russian surplus, but I have not even broken in the new barrel it came with. 
If the CMP still had them, the 1903's are capable of 1-2 moa.
Scmidt-Rubin's are supposed to be capable of 2 moa or better, but the ammo is expensive.

Buy you a couple of Mosins and put some high quality ammo through it.  See which one shoots best and sell the rest.

Really the best thing to do is get an LTR, ensure you can shoot 4moa or better, all the while doing that, getting as many people to Appleseeds so that they can learn the same skills and their American Heritage.

A Rifleman is capable out to 500 yards, with a standard rifle and surplus ammo.  Beyond that is beyond Appleseed.  If one's goal is beyond Rifleman's standard or beyond Appleseed, then they will be much better served seeking advice at places that specialize in that. 

The "best rifle" is the rifle you have or that you can get your hands on, cause it is the nut behind the butt pad that makes the difference.