The 1890s Mausers were very safe guns, for their own days and still true today. That's why nearly every modern bolt action owes at least some credit to Paul Mauser's design. Some guns, like the American Springfield 1903 and most American hunting rifles are nearly direct copies of the Mauser bolt system.
The safety rule of thumb is that you get the gun headspaced using the correct caliber gauges. If the headspace is too small, closing the chamber on a round becomes difficult. Too large, the real problem, and the case will split open and spew hot burning powder back at you. In extreme, and rare, cases, the bolt will break and travel backwards with great force.....and remember where your skull is!
Now I have shot many an old warhorses and I only sometimes check headspace, usually on Russian Mosins, as the ammo is hot and the bolt is not as safe as the Mauser.
When shooting an unknown gun, I will put in on bags on the bench, cover the action with a heavy piece of tarp I carry, about the size of a hand towel. I line the gun up downrange, turn my head and fire. Then inspect the spent shell carefully for any signs of bulging or cracks. Do this at least four or five times and if no cracks or bulges, you should be safe. I have used this method on many old rifles and only one, a Turk Mauser with a very worn bolt, showed any case cracking and it was minor. A different bolt solved that problem and it works great now.
My dad also has an Argentine Mauser he ordered from the Sears catalog. It was a beauty, think he paid $19 for it. We only shot it for fun, as he was going to make a hunter from it but it was just too pretty. That one was the start of my milsurp collection which now, 40 years later, represents nearly every country on six continents and two world wars.
The Argentine I would not worry about, just make sure the bore is clean and the bolt locks properly. The K98k is a bit different.
The Soviets captured millions of these during WWII and unlike the UK and US, they kept theirs in warehouses for a 'rainy day'. You see, the Russians had the unfortunate position of more soldiers than rifles when the Germans invaded and while it was very limited, men were really sent into battle without a weapon with orders to find one on the battlefield. They kept the German guns so they would have reserves. Before they put them in boxes, they took them apart and cleaned and blued (really black) them all usually including the bolts. They paid no attention putting them back together by serial number, so all are mis-matched. So the bolt is not the gun's actual bolt. Check the bolt action and lock up very carefully, both empty and with the chamber loaded and use the method I described for safety.
Now, most folks believe the modern hunting ammo is hotter than the surplus but that's not true in 8mm. American 8m huning ammo is very light and the surplus, especially Yugo and Turk, is very, very hot ammo that pushes the pressure limit of the gun. Although not cheap, I'd start with a box of Winchestewr or Remington in the 8mm, then work up to surplus fmj. You will feel the difference.
Also, if you set your target at under 100 yards, the sights will be off. Lowest sight setting on these is closer to 200 meters, so don't be afraid it doesn't shoot straight, you are just probably trying too close. Use a big target or if you can shoot at 200.
Good luck, let me know if you have more questions.