Remington 700 actions are relatively easy to come by, and are the basic stock many custom gun s are built from. The barrel manufacturers are plentiful. You have already been given some good names. Let me add Dan Lilja to it, also. My advice to you is to consult a GOOD gunsmith with your plan, and after discussing what you want with him, get a cost for him to install the barrel of your choice in the caliber of your choice on the action of your choice. As I stated earlier, the Rem. actions are a real good choice to build on. You should tell him you want the barreled action trued and the lugs lapped as a bare minimum. Most stock manufacturers, particularly the synthetic ones, are FAIRLY easy to work with in regards to getting the barreled action into. Wood stocks for the most part require a whole lot of fitting and handwork, and require the specialty tools to do it with. In any regard, you will highly likely be faced with having to glass-bed the action into either type of stock. Some manufacturers advertise their products as "drop-ins", but usually only for factory barreled actions. Your custom barrel will most likely require hand fitting in either case.
Building a custom rifle is a lot of fun, but it is very tedious work for the most part. Specialized tools are normally required, and a good basic understanding of steel-to-stock relationships.
I certainly would not try to discourage you from attempting this project. I only wish to alert you to SOME of the intracasies you will encounter.
The VERY first thing I would recommend you do is locate and talk to a qualified gunsmith and get a ballpark idea of what you are looking at.
Another option you may wish to explore is having a custom rifle built FOR you, with your input in the project.
Whatever you decide, good luck, and hang in there.