Dear Tony,
Just a warning, in case you don't already know, that after you do thorough research on this project, and you have priced all parts, gunsmithing and other things you need for your project, the cost of your project is going to be very high compared to buying a new medium grade commercial rifle.
This is probably OK, so long as you know it in advance and are just happy with the hobby.
For my money, I would just by a brand new or used Savage Scout Rifle, in .308 Winchester or 7mm-08. It is a very nice rendition of the scout concept for a reasonable price, in modern ammo, and very accurate.
Aside from that, I believe that the suggestions of the Enfield No. 4 Rifle (mark 4?) with its integral peep sight, or a Springfield 03-A3 that has already been sporterized (and still has its integral peep sight) is definitely the way to go.
The absolute cheapest way to go would be to buy an old Mauser 98 in good condition in 8mm, and then buy one of the pre-fab "scout rifle" kits that are sold for it, that provide a scope mounting sleeve that slides over the barrel and down to about two inches in front of the receiver, and is held in place with accra-glass and a locking screw. The problem is, even after you shorten and recrown the barrel, you still have a pretty heavy rifle, which is contrary to the scout concept.
My advice? Take a deep breath and think about all of this, and think about whether you are going to be happy with the end result after spending alot of money. And then, . . . go buy the Savage Scout in .308 Winchester. You will be able to use the relatively cheap 7.62 Nato ammo in it.
Just my opinion.
Big Paulie