Yes its possible, I have built several. When I lived in Alaska I bought about 7 Win M-70 actions from a guy. Most I turned into target rifles (300 WM 1000 yard rifles or a replica of a Vietnam era Mod 70 Sniper Rifle in 30-06. But target rifles are a bit differant so I'll mention my wifes 243.
I used the standard action, douglas barrel from Brownells, a light taper so she can carry it, cut to 22 inches. Lot of people like fiberglass stocks, I don't I like wood stocks. So people take their rifles to gunsmiths and have them re-stocked, so you can get new factory wood stocks fairly reasonable. Taking into account I had the machinery and already had the reamer & HS Gages, and blueing tanks, I think I have less then $100 bucks in this gun, most of that is the barrel.
The main reason I like building the rifles, is I like to build the rifle around the bullet and the purpose I want to shoot, instead of taking a rifle and trying to make do.
You can get 98 Mauser actions pretty cheap, and they make excellant rifles. You can get pre-chambered mauser barrels from Midway and Brownells, (you do have to do a final chamber fit for your
. You can get Fiberglass or Wood stocks pretty reasonable. Last nights Guns and Ammo show, they had a gun tip from Midway on final chamber fit. Its not that hard if you use care. Not only can you make a very nice rifle cheap, ITS YOUR RIFLE, YOU BUILT IT.
I know this is a simplfied answer and no doubt we're gonna get flooded with post saying I'm full of poop, I'll wait tell they come and I'll respond.
I'm just a firm believer that one can do anything if he/she wants too and puts their mind to it.