Hold on!
Not such a good idea, Buck! It would require a bunch of work, including opening up the bolt face to the .472 or so for the .30-06 and .308 case head dimension, and then you have to ream the chamber to the new cartridge. A 6.5-08 would be easier, close to the 51mm of the original 6.5 Carcano, but I don't know if there's enough meat to comfortably support the fatter .308 round after rechambering. A 6.5-06 would be much harder to do, being 64 mm long, etc.
Then you have to consider if that Carcano action is strong enough to handle the pressures of the 6.5-08 and 6.5-06. There's enough folks out there afraid of the striker assembly coming back into your eye even in the original chambering to seriously question wanting to open up that action for a hotter round like the 6.5-08. Not that I've ever seen or heard anybody come out with concrete evidence that the striker/eye accident really did happen with some frequency, though.
Matter of fact, there were a bunch of late-war Carcanos chambered and rebored to 8x57 Mauser. As an impromptu test, I helped to purposely blow one up by stuffing an 8mm Mauser case full of WW231 and a 170gr 8mm roundnose bullet into the chamber. The bolt stayed in battery, but the front ring let go, sending the barrel forward. Granted, it was much more oomph than the 6.5-08 or 6.5-06 would provide, but I wanted to see if that striker would budge.
Truthfully, if you want a 6.5-08 (aka .260 Remington)or 6.5-06, build one on a proven 98 Mauser action, or a newer Remington, Winchester, Sako, or Ruger action, it would be better dimensioned, and have a better safety margin.