Very beautiful and cozy looking space. I’ve found that unless the rug is just the right size the chair legs can catch, so the rug would need to be quite large. I think why you feel you need a rug under the table might be that, that side of the room feels more open. I’d try taking the rug away from the living room side to open up that side and see if that balances the areas out.

I agree - not a fan of a rug under a dining table. If you do want to have a rug, use a flat weave one so you can easily vacuum the crumbs and choose one large enough so that chairs when pulled out still have all the legs on the rug

Very nice use of plants, Holly N! I personally prefer rugs to hardwood and I'd put one there. Something creamy in the indoor/outdoor vein won't be too expensive and will be easy to clean. Or something natural (jute, sisal, etc) with a border that matches your sofa or other rug.

Your space is styled beautifully and I love your paint color. I think your dining room just looks a little barren compared to the coziness of your living room. I think replacing the rectangular mirror with art, curtains on the ends of the sliders, and something colorful on the table (colored glass vases, colorful patterned chair cushions?) would help more than adding a rug would.
If I had an unlimited budget I'd replace the wood chairs with upholstered slipcovered ones in a similar color to the couch. To help the table feel more balanced I'd put 2 chairs on the long side closest to the wall, and a bench on the long side closest to the couch.

I think rugs in the dining room, while they look nice, are more trouble than they're worth.

Your dining room pendant looks like it needs to be lowered 8 - 10".

Yes, you need a rug under the dining table.

I've always had a rug under the dining room table. This is one room that really should look welcoming and friendly, and a large rug in a similar tone and style as your living room one would look well. I agree about a painting instead of the mirror to inject more warmth and color. I would remove one of the floor mats or get a larger one. Two don't look very good. I'm afraid there's quite a disconnect between the barn door window covering and the ladder vis a vis the sleek and modern furniture and elegant rug. They also make that side of the living room look busy and crowded. Objects that are trendy don't automatically fit into every decor. I would suggest whitewashing the fireplace to make the room look larger and more in sync with your beautiful furniture. Your living room rug is a bit on the small side. Ideally the chairs should be pulled closer and also sit on the rug to make the scheme look united and cohesive.

If no rug, put velvet stickers under bottom of chair legs to prevent scratching floor.

Measure your table and add 36-45" to that size all the way around. That's the size rug you need and it shouldn't run into walls, furniture or baseboards. Doubt you have room for one.

We've designed plenty of dining spaces without rugs under them; some when they were next to the living room: