Rummer: this is just from my personal perspective, but I wonder why you would want to 'download' the 8mm by placing it in a shorter case. The 308 is a 7.62x51mm round - the 8mm is a 8x57mm round.
I do not know what the original caliber of your Rem 700 was but, if it is a long action and you are looking at an 8mm bore, I would just go with the 8x57 or a longer 8mm even. If the Rem is a short action it may still be able to handle the 57 mm length cartridge, but I would check with a gunsmith on that one or possibly even the Remington factory (they are in NY, in Ilion, in the 315 area code). Oh, BTW, The 2004 Remington Classic is chambered for the 8x57. I wonder how that action compares to yours???? HTH. Mikey.