223 and 308 only denote the bullet diameter, not the cartridge they are being used in or the rifle they are being fired from.
Now, BAT, Noseka, and several others make good target receivers. Lilja, Shilen, McGowen, Kreeger and several others make good target barrels. McMillen and Lone Wolf make good target stocks. Nightforce, Leupold and a few others make good optics. Norma and Lapua make good brass. Berger, Lost River, Nosler, Sierra and a couple others make good bullets. BUT, without a great gunsmith to assemble the package with care, not even the best collection of parts will proform at it's potental.
On the flip side, you could start with a Rem rifle, have the receiver trued and bluprinted (after which the origional barrel won't fit anymore), replace the barrel and stock and you'll still end up spending about the same amount of money as if you'd started with a high end, but ready to build, receiver.
Of course if all your wanting to do is play at poking holes in paper, than a Salvage or Dumchester bolt gun would work just fine