I use the Heavy Duty Felt Pads on my chairs that are on the hardwood floor. You can buy them at Bed, Bath & Beyond or Home Depot. I just change them every once in a while but the chairs slide easily and no scratching

Do your chairs have the nylon glides they came with?

There are after market glides with felt bottoms.

https://www.shifflerequip.com/floor-protection-sled-base-glides.aspx

Indoor / outdoor rug where the carpet loops are level.

Ballard Designs always has some nice rugs for indoor outdoor.

I use felt pads under my chairs so they do not scratch my light porcelain tiles. They work very well, are cheap and can be easily replaced. After 4 years I only have to replace them now on some of the chairs. They stick on and are especially made for chairs. I do not like rugs under a dining table. I would go mad with the thought of drinks and food falling on my rug, especially if it is a natural material.

You could check out Chilewich floor mats and BTW those felt pads will not work on those chairs

A Flatweave will be the best rug option. Chairs won't necessarily "Slide" but will be easier to move around on a shorter nap. Here's a great starting point: https://kanecarpet.com/collection-name/flatweave-collection/

Thanks all. I'm not a big fan of dining room rugs either but our landlord requires a rug in this room unfortunately. I appreciate the suggestions - I will check out Chilewich and Ballard...and will look for more flatweaves although they are hard to shop for online. Thanks all!

HA! A landlord can't require an area rug, any more than he can require a picture on a wall. Not unless he has provided, same and made it impossible to remove. Just noticed that "gem" : )

He's living in there with you? Or he lives below ?

Jan, a landlord can certainly charge the renter for damage to hardwood floor ... specially after he requires them to use an area rug under furniture! I'm assuming the request is included in the lease. The OP chose to rent the place with the unusual requirement.