I was going to post earlier, but I see that Tiangou got there before me.
The finish on the table is failing. Your best bet is to strip it off, carefully of course since it is a wood laminate surface, stain if needed, and apply a modern finish such as a polyurethane. As the article says, match the finish to the intended use for best results.
I have an old rocking chair that needs done. I had assumed that the areas that were failing, since they were where one would put one's hands when sitting in the chair, were simply dirty from oils from the hands. I finally got them cleaned up (i.e., no longer sticky) only to find that I had mostly stripped the finish in those areas!