If the peaches at hand aren't fully ripe yet, you can store them safely without any danger of them getting spoilt. The best thing to do in such a situation is to just leave the peaches on a countertop at room temperate for a few days until they're good and ready. If you want to speed up the process, throw the peaches in a plastic bag with a banana, and you'll have ripe fruit in just a couple of days.
If the peaches you buy are ripe and ready to eat, it's best to just gobble them up as soon as you can. If you do need somewhere to stash them for a couple of days, you could refrigerate the peaches. However, since peaches are always ripening, they will keep 'cooking' in your fridge, too, just at a shorter pace. Refrigerating your peaches for a long time will dehydrate them and make them all mushy.