The action is too short for the 8mm Rem mag. For what it would cost you to have the action lengthened (and weakened in theprocess) you could buy a brand new 8mm magnum.
There are other chambering options, however.
(1) 8mm/.338 Win mag or
(2) 8mm/350 Remington Mag.
The 8mm/350 Rem Mag is a 308 length case that would feed like a champ in the 57 mm length magazine of your Turk. In fact it would allow longer seating of the bullets for more powder in the case. You would only have to rechamber, alter the bolt face and alter the magazine.
I would not advise the 8mm-06. Too much work for too little gain, frankly. A lot of hassle to handload a round that's not any better than a handloaded 8x57 and with the 8x57 you can still shoot cheap surplus ammo.
Before doing anything, shoot it as is to establish the accuracy level of the barrel. If it won't group well as an 8x57, it probably won't do any better with rechambering. Beware the lure of "sportserizing" military rifles. You can really pay out a lot of money and end up with a pile of junk. I speak from experience. Frankly if you really need something as powerful as an 8mm magnum, you probably need a better rifle than a surplus Turkish mauser.