Dogdinger, for a straight rebarrel, I think the labor cost would be $350 - $500, plus whatever you worked out with the 'smith for reamer and headspace gauge cost. If he has both, there might not be a charge. If he has to buy both, anywhere from you paying all the cost and you own the reamer, to splitting the cost and the 'smith keeps the reamer and gauge. The rebarrel should include truing the receiver face and lapping the locking lugs. Additional "blueprinting" would be extra. A Remington 700 rebarrel for instance would be at the bottom end of the cost range, and a Pre '64 or classic mod 70 Win, toward the upper end. Also, a falling block such as a Ruger #1, Browning 1885 or B78 would be at the upper end of the range too. Other common additional costs would be having to "clock" a barrel to the action, and/or installing a brake.
There may also be some FFL and shipping costs. The 'smith will need the complete action, not just the barrel.
These are ballpark estimates. Different 'smiths have widely varying prices.
The SAAMI max OAL for the .243 Win is 2.710". The SAAMI max OAL for the 6mm-284 is about 1/10" longer. It's also fairly common to seat longer bullets further out in the 6mm-284, so be sure whatever action you use will handle some extra length if you want to feed from the magazine. Both cartridges have .473" base diameters, so that works fine.
Good luck, Tom