Hi Steve, glad to have you here. You sound quite young, I'm guessing in your early to mid teens, this is a good thing glad see young people with as much interest as you in the sport.
The simple fact of the matter is scratch building an action is only worth it if you have a lot of money and time and are doing it merely as a hobby. There is no way to make your scratch built action more economical than a commercial one, in my mind. You will be shooting much much sooner with a commercial action as well.
As for actions alone, unless you are planning to buy a custom action (much more $ than a complete commercial rifle, but also much better) it is cheaper to buy the complete rifle and take the action from it.
Next, .338 Lapua has heavy recoil without a muzzle brake, very heavy. It is not pleasant to shoot without a brake unless you are using a 20lb rifle, you will more than likely develop a flinch from it. I have owned up to .375 RUM in an 8lb rifle, so I know recoil and am comfortable with it, but I wouldn't want to do unbraked shooting from a Lapua for long. In a standard weight rifle without a brake, recoil is downright
brutal.
Next, there's no such thing as a "good/cheap barrel", spend your money wisely. You get one or the other, good or cheap, they seldom come together. You mentioned buying US made barrels, do realize you need export permits and paperwork to take a barrel out of the US and this is time consuming and pricey.
Next, there's no need to go .338, a 7mm as mysticplayer adeptly suggested is a far sounder choice for your applications. The 7mm VLD's have similar ballistic efficiency, and the cartridges less recoil, and burn less powder. They also fit in a standard magnum action and feed without time consuming and possibly expensive modification, unlike the .338 Lapua. You will get to shoot more as everything's cheaper, there's no way you'll make it to becomming an adept long range shot on shooting 5 rounds every once in awhile when you can afford it.
Also note that while you wish to shoot very far with this rifle, the scopes you have mentioned will limit your range. You will need to go to the Mark 4 Long Range series of Leupolds ($1200-$2500) to get the elevation range required for very long shooting. Otherwise, your .338 will just have the reach of any other magnum really.
I had numerous plans and schemes when younger, still do.
I just had to decide when they were too much, I do believe this is a case. That said, you will likely build you rifle, and I look forward to hearing what you come up with, I think a Stevens 200 ($329) in 7mm Magnum shooting VLD's is a prime choice. The entire rifle will cost merely a portion of a good, long .338 barrel.
