As cheap as possible and .338 LM don't go well together...
I don't mean to be a d!ck, but you may want to rethink getting a .338 if this is your thought process. I worked in firearms retail for several years, and I saw numerous guys buy .338's and then dump them 6 months later because they were "too expensive" to shoot. I'm not trying to convince you not to buy/build one, I just want to make sure that you know it's going to get expensive... both to buy/build and to shoot, and there aren't too many places where you can make full use of it. Plus, I would mention that buying all the parts separately, even if you're assembling it yourself (I have no idea if you've done something like that before, but I would not recommend it unless you have extensive gunsmithing experience), will likely be almost as expensive, or even more expensive than buying a factory rifle (You'll likely be looking at $500 for a stripped action, $500 for a barrel blank, Several hundred for a stock, a few hundred more for all of the other misc. parts, then assembly time, chambering the barrel blank, profiling and crowning the barrel, etc... Remington 700's in .338 are available from Wolverine at a little over $2k, plus optics, plus everything else).