Sandpaper, true oil and a $50 checkering tool set. You can do it yourself.
I will add a caveat to that. I have used Tru-Oil a couple times. I NEVER use it straight up. I always cut it at least 40% with Paint Thinner prior to use and then I use a wet sand method to apply it. Once the wood has been sanded to at least 400 grit (I prefer 600), Apply a thin coat of tru oil with the tip of your finger (prefereably with gloves on) and then sand it in with a piece of 600-800 grit wet/dry sand paper. I sand until no more liquid is present and it has become a sort of wood/thinned Tru-Oil slurry. Then I let it sit for 15 minutes, then I buff it off then let it sit overnight. It will take about 10-12 coats like this to properly fill in the grain. Then, I apply a coat, let it sit for 15 minutes, buff it off and let it sit overnight. No sanding after grain is filled, there is no need. I do about 4-5 coats like this with the thinned out Tru-Oil. Then I let it sit for 3-5 days to fully set. After it is fully cured, I polish it off in the direction of the grain with ultra fine buffing wheel polish applied to a cotton rag (you may have to melt the polish a bit to get it to adhere to the rag.) ONLY go in the direction of the grain if you don't want swirl marks to show up. Then apply a paste wax designed for furniture.Agree with 90% of what @ChrisG said. TruOil is really shiny stuff because it has some solids in it that are less than "oil". It is easier to use, hardens faster, and creates that "Perazzi" mirror finish. It also lets you do a job in 4-10 coats whereas a real oil finish might take 20-50 coats. (and the word coat is a misnomer, is a drop on the palm of the hand rubbed in every day for two months. (Trivia: Purdey's technique took >90 days. A coat a day for a week. A coat a week for a month. A coat a month for 3 months.) It also is a very modern look ala Roy Weatherby gloss rifles. I've tried to use TruOil and then to knock down the gloss on the final polish a bit. After a couple weeks of shooting the shotgun it burnished to gloss again wherever my hand and thumb were located making it look a bit wonky on a gun made in 1896. (A Lancaster Sidelock Single Trigger 12 bore)
There are finish kits out there that include the red alkanet root stain (or use vinegar and iron shavings if you want the alternative), slacum, and proper grain filler. If you want to restore it, you want to reapply the traditional finish techniques that will give you the right look. It appears from the photos its a German trade gun like a Simpson, a Sauer, or some other guild gun. They would have used a more subtle, subdued oil finish originally.