It's important to let the peaches condition at room temperature before you store them in the refrigerator.
You know your peaches are done conditioning if they give a little when lightly squeezed. At that point, you can store them in the fridge for about a week. Refrigeration stops the conditioning process, so the cold temperature will maintain the degree of softness and juiciness your peaches have when you put them in.
Most grocery store peaches miss the mark because they sit in refrigeration for far too long and never condition properly.
Remember that each peach is its own little gem. Some peaches may be ready to go in the fridge (or be eaten!) while others remain firm. Be patient with the stragglers.