Here is a list of average components cost. You can go way above if you want.
Custom action $1000-1400 (Stiller, Defiance, BAT, etc)
Barrel 1.350 at 28-32" $500 (Hart, Bartlein, Brux, Krieger)
Trigger (Basix to Jewell) $125-225
40 MOA Base $100-150
Rings $80-150
Muzzlebrake installed $200-250 Holland Radial is very good for the money in particular.
Chambering $185-$400
Bedding $200
Stock $500-1000 (McM, Russo, Manners)
Misc $200-300 (ltrigger guard, recoil lug, sling studs, flush cups, recoil pad? etc)
Bottom metal & magazine $250-400, (IMO for hunting waste of money)
Bottom line lower end is right at $3000 with no scope, brass or dies to tops of whatever the smith thinks he can get in some cases and they are real proud of their when you break it out like this. They will not give it broken out, rather "package price" which is rarely cheaper. That is also assuming not an improved version with custom dies that will be $200-400
You have two options to minimize costs.
1. Watch here and Snipers Hide for good used 338. Lot of them available.
2. Start assembling parts as you can find them. Be careful of the barrel specs as you will want a min of 1.250 IF not more on the shank. If using a custom action 1.350 straight for 5 inches to allow one setback. Min length of 28 to max of probably 32.
IMO custom action is the ONLY way to go. No work, usually readily available. Stiller and Defiance right now two of the best for delivery and money.
BH