The 223 meets your requirements, except for current ammo availability. Many areas don't have any on the shelf.
If you already handload that makes many cartridges close to the 223 price. If you don't handload you should. It reduces the cost of shooting by enough that it pays for itself quickly if you shoot often. If you don't handload the other military surplus cartridges will work. Some of the surplus rifles would work, however accuracy is a craps shoot. New rifles are chambered in 308 win (milsurp ammo but not as cheap as 223).
Many bolt action sporter weight barrels are available for $ 5-600. A few to look at are Ruger American, Marlin XL7, Savage (I can't remember the model, someone else will likely mention it). Almost any bolt action will get you to 300 yards.
You can't fix stupid....however ignorance can be cured through education!