$250 gives you a few options with just the barrel. Consider the fact that you would spend $30-$50 on a gas block and another $10 on a gas tube PLUS separate shipping charges if you order them from different places, and it adds up quick. Keep that in mind with these options below.
White Oak (with gas block) - $285 (very likely a long wait on this though, WORTH IT)
White Oak Armament | AR15 Barrels | 20 inch .223 Varmint Barrel | 1 in 7 Twist with fitted Gas Manifold
Wilson Combat (no gas block) - $275 (another long backorder, but very worth it IMO)
Super Sniper Barrel | .223 Wylde | 20" | Stainless-Wilson Combat
Rock River Arms - $290 to $310 (I've heard very good things about these, but your handguard choices will be limited due to the bump in diameter after the gas block)
Rock River Arms: Stainless Steel Varmint Barrels
Yankee Hill Machine - $240 (one of the only non-stainless ones I've ever seen)
20" Fluted Melonite QPQ Barrels
Value Pick: DPMS Bull Barrel: $201
DPMS Barrel AR-15 223 Remington .920 Muzzle Diameter 1 9 Twist 20 SS
Be advised: whatever you do, make sure you find a handguard with a barrel nut that will slide over the heavy profile past the gas block. I had to have a barrel nut turned from 1.005" to 1.055" on the inside diameter because it wasn't big enough to slide over the shoulder past the gas block.
Build your gun around the barrel, any one of those manufacturers (especially the top tier) will have good headspacing and with a Mil-Spec bolt or similar (think the AIM surplus or the Ares Armor BCG's I linked you in another thread), you shouldn't have headspace issues.