Find a different rifle if you want a 7-08. You are taking the hard route, believe me.
This is why you see [bleep] up rifles in some pawn shop, the result of poor planning and a blacksmith hack job.
MtnHtr
Ditto on Mtn Hunter's points. I looked into a similar mod with an old M700 of my dad's, also in .223. Part of my rationale for my project was that my dad passed away and I wanted to shoot a bigger round,
but in his gun. Whatever.
Besides having problems opening the bolt face to .473" to accomodate the 7mm-08, you'd run into much bigger feeding problems.
DON'T DO IT!!!!Remington short-actions (in M700's, 721's, 600's, Model 7's, etc.) are scarce items! Whether an action has a .22-250/7mm-08/.308Win-size bolt, or a .223Rem bolt - there are shooters looking for rifles just for their actions who'll pay dearly to get the action for a project-rifle.
Off hand, if your rifle was in like-new condition it would be worth at least $400 to someone wanting just the action.
If you want a 7mm-08 in a M7, buy a Model Seven already in that cartridge - or find a .308 or .243 M7 (for a song!) and have it rebarreled.
Brownell's sells brand new M700 actions for $400 or so.