The first question is which .338 do you want to build? The answer to that question will go a long way toward deciding what action to use as a base action.
The second question is what is your budget?
A good third question to ask yourself is how much you want your rifle to weigh. That will control component selection and barrel contour and may have a large impact on whether or not you want to use a muzzle brake
Another question to ask yourself is what bullets do you want to shoot? The answer to that question may have an impact on the twist rate you choose.
The budget question will go a long way toward deciding which barrel maker and style to go with.
Though I am not personally a fan of them, a Remington 700 long action, or one of its custom clones, will give you enough length to accomodate 338 RUM length cartridges or a 340 Weatherby.
338 Lapua is available in both Remington and Savage actions, but there are those who say the cartridge case diameter of the Lapua is pushing the safety margins a bit in both actions. There are custom actions that are purpose built for the Lapua, however. The Stiller TAC 338 is a good example of that.
If you are after a 338-378, a Weatherby Mark V of the appropriate action size would be a good choice.
If you want to build a 338 WM, you have a pretty broad choice of actions.
If you want to use a factory rifle as a base gun for a build, I would recommend that you make sure a Wyatt's Extended Mag Box is available for the action you are considering. You are likely to want the extra OAL to get the best performance from the high bc bullets available in 338.
Where the barrel is concerned, price and desired twist rate will have the most effect on your choice. Desired rifling type also plays into this as well. Button rifled barrels tend to be less expensive. Cut rifled barrels tend to be more expensive.
If you don't have a big budget to build your 338, there are some pretty good factory options out there. For instance, I am not normally a Savage guy, but the 111 Long Range Hunter in 338 Lapua definitely has my attention. It may very well go on my wish list.