First, stain does not work if there is a finish on the wood---which is present on that table.

The easiest and most durable finish for the top will be non poly varnish. The easiest way to apply that is to use a wipe on mix. It takes six or more coats of wipe on varnish to equal three coats of brushed or sprayed on varnish, but for a DIYer, wiping is almost foolproof.

Likewise, you shouldn't really put an oil finish over an film-forming finish. The most likely finish on there now is lacquer. That said, there are many finishes that on the label would appear to be "oil" such as tung oil finish, antique oil finish, etc., that are really wiping varnishes.

You should also do a thorough cleaning to get rid of accumulated grease, body oil, foods, and general soiling before attempting any finish over the top.

Another complication is that if there is silicone oil from Pledge on there. That can cause many finishes to fish eye (crater).

Thanks! After cleaning, should I strip it then? Or just sand really well? Or just paint it with the chalk paint? Lol.

The seller said it was at least 50 years old. It belonged to her patents. Just in case that's any indication of what the current finish could be.

If you want to change the color, it's probably best to do the stripping. (There are other ways, but they're not particularly DIY level). Don't just "sand off the old finish." Worst case, you zip through the veneer to the substrate, best case, you don't completely remove the finish or remove it evenly and your stain will be uneven.

The discussion above led me to believe you just wanted to top coat. If you want something different, then yes, strip.

Lacquer became used after WWI and displace shellac for the next 20 years. After WWII, almost all finishes are lacquer up until recently when there are a few other things coming in.

Thanks so much Bobsmyuncle! Yes, I really just wanted a top coat. One, particularly that might bring out the richness of the wood, but a top coat nonetheless. What do I use to get the old stuff off then?