Well, here's the real question: what kind of accuracy do you require and what kind of sights do you feel you will need? If you really want 1-2" at 200 yards, then cheap ammo is not part of the equation. For that you'll have to go with handloads, good commercial loads or "sniper/match" grade surplus, and of those only the handloads are cheap.
Now getting to 200 yards with reasonable drop isn't much of a problem. All you need is a rifle shooting at 1800 fps or so and pretty much all modern centerfires do that.
7.62x39 or 7.62x54R are about the cheapest out there and there are some decent loads available for them. I find 7.62x39 Golden Tiger to be fairly accurate. I can get 2 MOA groups with it using my Saiga if I allow the barrel to cool between shots. Unfortunately, the only tackdriving rifle I'm aware of in 7.62x39 is the CZ527 and that will run you about $500 and then you'll want to add glass. In 7.62x54R, 148 gr. Wolf shoots really well out of my M39. I get about 2 MOA with it and that's with iron sights. Unfortunately, Mosin Nagant's don't really lend themselves to scope mounting and the Russian ones seem to have pretty lackluster accuracy.
5.45x39 is also cheap, but you'll pretty much be limited to shooting it out of a Romanian AK.
Cheap commercial plinking ammo for the .223 and .308 tend to deliver abysmal accuracy.
So if you really want to shoot cheaply and accurately, you can buy a used Remchester with a decent scope and then spend another $4-500 on handloading equipment and supplies. With some work that should get you to MOA accuracy for around 30 cents a shot and will cost you maybe $900.
Another option might be to find a good shooting Mosin Nagant and scope it. Most aren't terribly accurate, so you might need to go through half a dozen or so before you find a really good one. Or you could spend $300 and get a good Finnish rifle, (those are accurate). Add a scout mount, pistol scope and a lace up cheekpiece and you'll have a 2 MOA rifle using 50 cent ammo. You can still shoot the cheap surplus stuff, but don't expect the same accuracy with it. With handloads you might even get it down to MOA.