The first step is to identify what finish is on the table now.

It may be shellac and easily restored by applying some new layers.

How do I tell? It almost seems like it is just stained because the sheen is only present after I oil it.

This came from an antique shop in the Washington DC area and was purchased about 40 years ago. It was deemed an antique even then. I believe the wood may be walnut.

I am attaching a pic of it.

Use this preparation to clean, and restore wood furniture.

In the top of an old double boiler or in a two-quart can, add:
3 tablespoons boiled linseed oil (buy it as such)
2 tablespoons turpentine
1 quart hot water
Place over hot water until the moisture is as warm as a gloved hand can stand. For best
results, provide a means for keeping the solution hot while it is being used.
Dip in a soft, clean cloth into the solution. Lift it out and squeeze until it does not drip.
Go over an area no larger than about 1 by 2 feet, redipping the cloth in the hot solution,
squeezing it until it does not drip, and washing until the area is clean. Follow immediately with a
clean, dry, absorbent cloth until all oil is removed. Boiled linseed oil allowed to remain on the
wood will become tacky. Continue until the entire piece has been cleaned.
The solution not only cleans woods, it also protects the finish. The hot water and
turpentine removes the soil and the boiled linseed oil lubricates the wood, thus preventing
hairline checks in the finish. It also makes checks already in the wood less noticeable.

Thank you Sloyd. I will see if I can get my FIL to help me.