Its not only doable, its practical.
Hang around gun shops, pawn shops, and find a beat up Remington 700. The more beat up the better (because it cheaper and actions are hard to scew up).
I had an old Rem 700 action that's been kicking around for a while. Don't know where I got it, some sort of trade or did some work for some one.
I desided to make an Across the Course Rifle for high power. So I built a 26 in 308, good target stock. Iron sights, the works, only it wouldnt shoot for crap.
Fast forward 20 years or so. Granddaughter needed a science project which consisted of cutting the barrel from 26 to 16 inches two inches at a time. (I've wrote about this project before).
So I'm left with a Remington action, and a grandson who was turning 12, old enough to hunt in Wyoming.
So I get a stainless barrel. about 100 bucks, I get a fiberglas stock for less then 100. and comence to building my grandson a Christmas present.
I had the reamer and headspace gages, (you can rent those). I bought a Weaver K-4 ($135), and start shooting.
It's an excellent shooter, under 2 MOA (its a hunting rifle) for a grand total of less then $335.
Smile on grandson's face come Christmas morning was worth that 100 fold.
NOTE: If you can't find a Remington action in your price range, think about a Mauser action. They can be had for less then $100 and make damn nice rifles.