That must be a wooden, non checkered stock.
Just sand and get all the old stuff off. When you think you are done go over it a couple more times.
Use Tru-oil from Wal-Mart or most better gun shops.
Apply about a scant teaspoonful and rub around w/ your hand-no rags or paper towels needed. Allow to dry-takes overnight most likely. Buff off w/ steel wool-00 or 0000, repeat for several coats-till all pores (pits) are covered.
You will very quickly see that unbuffed TO is HIGH gloss but buffed is really nice matte finish. Just keep applying and you will soon be able to tell when enough is enough.
Many others will recommend a variety of prods, oils, minwaxes etc etc.
TO is tough, weather resistant and patchable when you scratch it. Only need to redo the affected area.
You will get compliaments.