Circa 1850, or my favorite gelled poly stripper, or any paint remover. "Take the stock off the gun first" butt and fore end. Make sure the butt plate is off the stock as some of these strippers will soften and eat plastic, that particular gun has a metal butt plate, no matter take it off anyways. Lather the stripper on, let it sit for 10 -15 min until the varnish bubbles up. Wipe it off with 000 steel wool. Repeat until you have it all off down to the wood. That stock appears light in colour for a 54 vintage. May have been sanded previously, not to worry. After getting the varnish off rub it down good with #0000 steel wool. If the wood is to light in colour for your liking you can stain it a darker. You have options here, either water based or oil based stains. Water based dry quickly and don't affect the drying of the oil finish. Oil based stains take longer to dry and may affect your oil finish drying rates and hardness if the stain is not left to dry up well. Patience and more patience is a good thing when refinishing a stock.
Sounds like you don't like gloss finishes or you wouldn't be tackling this. Walnut is an open grain wood and the pores need filled. There are many ways to do this. One easy and effective way is to lightly sand the wood with 220 grit, leave the dust on the wood and apply a drying oil such as polymerized tung (Lee valley Tools) several drops at a time to create a slury of wood dust and oil rub it all in good and let it dry for several days. Take it all off down to the woods surface with #0000. Add two more coats of the oil drying it well between coats. Take it down again to wood with #0000. The pores should be filled by now if not apply more oil and take off until it they are. then apply your final finish. Worse comes to worse and you don't like your results you can just take it down to the wood again and start over so relax take your time don't be afraid to experiment a little. Google and CGN is your friend for getting info and recipe's for finishes. Keep in mind there are as many recipe's for gun stock oil finishes as there are for chocolate chip cookie's. Personally I use several different types depending on the gun and what I want to achieve. Tung 100% pure, polymerized tung, (tung oil with dryers), Lin-Speed Oil. A commercial product linseed oil based, very good product and gives great results. Not available in Canada though. A lot of folks swear by Birchwood Casey's True Oil. It's okay, I have a tin of it, I was never overly impressed with the results. But a lot of guys swear by it. You have to leave any of these finishes dry up for at least a month or so after they feel dry to let them harden up. Even longer when your using tung oil without dryers. Stay away from boiled or raw linseed oil, in this state it has very poor abilities to resist water despite what your grandfather says. Hope this helps and gives you some ideas.