There are so many options it will make your head spin. You have more people making "sniper rifles" today than ever before. The market is completely flooded with them and everybody's is the latest greatest on the market.
I may over simplify things here, but…$4000 for some of these rifles are just way too much. Today's "out of the box" production rifles are very close and cost thousands less. There are a few companies out there (as previously mentioned) that will build your rifle on your action.
Get a good sound action, I prefer Remington but that is because I have been using them for many years and they "feel" very familiar. I keep trying others but this one works for my needs. So take your action, send it to someone like HS (as mentioned), they will barrel it, true it, lap it, stock it, etc. for around $1500. If you don't already have an action, you can buy a dirt cheap Remington 700 rifle (youth model) for around $400. Sell the stock to someone with a "youth" and get another $50 to add toward the "upgrade." This is the same process that the $4000 rifles go through except they supply the action and charge you the extra money.
Now, of course there are companies out there that build their own actions, and/or barrels, and complete rifles around those components. For something specialized you can and should expect to pay more. But for a supped up Remington, I can not see spending $4000 when I can get out the door for $2000 or less.
Shop around, there are a few companies offering this "upgrade" and the results are outstanding. We are getting benchrest accuracy out of our tactical rifles these days.