not without thoroughly sanding all the original finish off first. stain will not penetrate whatever varnish or sealer was used.

That table is in good condition. Get a tablecloth.

You don't need to sand the original finish off, but you do have to strip it down. Like B F said stain will not penetrate the sealer used, so you need to strip that off. I recently used an Eco friendly remover calledEZ Strip to remove paint and it worked pretty well.

I guess what I meant is that if the table was originally stained, you'll need to sand down to the virgin wood or it'll be blotchy.

staining is really hard though. the tiniest flaw will show badly. if you missed a spot removing the topcoat, or if you sand in different directions. all of these will show up as ugly streaks when the stain goes on.

don't get me started on what kinds of stain and topcoat to use. poly? water based? so many options, which yield very different results when it comes to looks, durability, smell (some finishes take about a month to stop smelling). it's also really hard to do a good job with the clearcoat to avoid brush strokes, dust and particles sticking to it. etc.

honestly, unless you know your stuff with DIY, I'd leave this project to a pro. or buy a new table.