Many, many years ago I bought my dad a Winchester 670 in .243. It had a 19-1/2" barrel, and belongs to my son now. IIRC it's about a 1.5 MOA shooter with typical .243 velocities. Not quite as good as my M 70 Varmint Grade, but plenty good enough for a lightweight hunting rifle. Might be able to get more out of it with a little work.
Also, to piggyback on Petrock's post, I have been building a .308 target/tactical rifle on a Remington 700 AAC SD Tactical platform. It comes witha 20" barrel, and produces 1/2 MOA groups on a regular basis, which is on par with my 24" barrel .243 varmint rifle.
In my experience, a .243 Winchester with a 20" barrel will absoulutely do what you want to do. True, that the muzzle blast will be, uh, noticable, but .243 is not a quiet round anyway, no matter how long the barrel is. I really don't notice the muzzle blast any more with the 19-1/2" barrel than the 24" barrel. You will get more flash if you fire it at dusk, though, so it may take your eyes a second to re-adjust.
I'm a big fan of the .243. You can go from ground hogs to deer, ethically, and won't be over or under-gunned for any game in that range.