If it was some type of varnish, it's more complicated. If it was an oil, like Watco, etc., easy; just apply a fresh coat of Watco after a thorough cleaning.
Linseed oil I wouldn't recommend. It doesn't dry very hard as a finish, and it darkens wood considerably. And the darkening is not reversible.
If varnish, it was probably an old-fashion oil type rather than a poly. Test to see if it was shellac first, by wiping with alcohol; if the finish comes off, that's a positive for shellac.
You can still find non-poly varnishes, in all sheen levels for interior finishes. Apply according to label directions after a thorough cleaning and sanding with 220 to scuff the surface.
Another (more foolproof) idea is a wipe-on varnish, which combines the ease of oil, but builds more of a surface film, and it can be applied over other varnishes.
Wipe-ons are either liquids or gels.
In any case, you will need to clean and prep.

Casey

Casey, thank you! Looks like a very sound and thorough map-out of how to proceed. I appreciate your warning about the darkening aspect, I don't want to ruin this beautiful wood.

Linseed oil pretty much ends up black after it finally hardens (over many months to years).

You can add driers to speed it up a little, but it is NOT a very good finish.

Seems like linseed oil is a bad idea here, thank you for pointing that out too. The salesperson at the local woodworker supply recommended linseed oil based on what I told him about the problem, but we didn't talk about darkening at all so I'm glad this discussion brought that out. I am still considering oil (would look for the Watco product Casey mentions) though wipe on varnish is looking better, can anyone tell me whether one darkens more than the other? Have read very good reviews of General Finishes products here on garden web (kitchens)--from their website, looks like their oil-based topcoat (as contrasted with their oil based stain/sealer products) which is called "GF Arm-R-Seal Urethane Topcoat" which is liquid wipe on, would be a good bet, or "GF Oil Based Urethane Topcoat" which is gel wipe on. Any thoughts on these specific products, or any recommendations for other products? I've used Benjamin Moore oil based clear stain in the past and liked it. I'm just trying to avoid sanding on this project, I've done that before on smoke damaged tongue and groove plank ceilings (Deck House again) and the results were good, but it's too time consuming and disruptive at this stage. I think I can manage a wipe-on finishing project, but not sanding. Thanks again!

Be sure and test first.

A new finish may not stick well to the old varnish without some prep work (like light sanding or even stripping).

Thanks much, appreciate your advice--will test it first.

hi eggpainter, to benefit other Deck House homeowners, could you please keep us posted what products and procedures that you end-up to use and how was the result ?