Can't be done on most sporting rifles. You can't set the barrel back far enough BEFORE you essentially run out of steel to chamber because of the distance involved are you are into the taper of MOST barrels and out of room. Only way it's going to work is if your rifle has a bull/untapered barrel hat leaves enough steel for the chamber. This isn't going to be an easy cut it back a thread or two jobs. Your going to have way more into it than the cost of brass, for no particular gain in accuracy or ballistically 7-08 = 7x57 284W = 280 Rem
If you can't find Winchester brass you can buy 6.5-284 brass from Nosler, Norma or Lapua and run it through a 284 die to expand the neck.
I shoot several cartridges based on the 284 Win, 6-284, 25-284, 6.5-284, 284 and 30-284 and while 284 brass availability has always been a bit spotty. The popularity of the 6.5-284 in BR circles pretty much solved the problem AND any one of the three is much better/more uniform, brass than the Winchester.
I built a #1 in 7-08 a couple of years ago and had LOTS more trouble finding 7-08 head stamped brass than 284 brass. Finally just made them out of 308 until some showed up.
As an aside I felt it was important that it be head stamped correctly. You can't fire a 7-08 in your 243 but you can fire it in your 308 if your not careful. Doesn't hurt anything I guess, as the bullet just sort of rattles down the barrel but it's a BIT disconcerting and accuracy suffers. ;-)
This is sort of like the 240 Weatherby there's simply no commercially available .243/6mm that can be chambered into a 240 Wby as it's smaller in diameter along it's entire length than they are and it's long so you simply can't set a 243 or 6mm Remington back far enough. Okay, you can do it with a 6mm PPC but you have to open up the bolt face on bolt guns AND ruin what is probably a fairly rare/expensive rifle.
RWT