Hi. What are the room's dimensions, and what kind of fridge? Do you have some in mind? I'm not sure where one would fit without knowing the room size and how big the table and rug set up is.

Depending on the room’s dimensions, a sideboard with built-in wine and beverage refrigeration may work in your dining room.

I measured the room. 10ft by 13ft

The table 85 inches (7ft)x 40 inches (3.5 ft)
Room is 120 inches (10ft)x 156 inches (13 ft)

Rug is 7ft x 10ft

A wine fridge will not be attractive in here. If you cannot afford a sideboard with built in fridge, purchase a deep console table to house it in. There must be someone on Etsy who makes such a thing. Hang nice artwork above it.

Have you tried the table going the other direction?

I noticed you have no chandelier. I think the room could use a fixture a ove the table.

Okay, first you want the furniture you have set up correctly for dining at the table.

You want to have 3' of space between the walls and the side of the table and the rug should be minimally 2' wider and longer than the table so that the back 2 legs don't fall off the edge of the rug when some one pulls away from the table. (30" is better, but not always possible.)

Your rug is a bit too small - 18" longer than the table on each end and 21" wider on each side. We sometimes have to make do with what we have, but if it is returnable or you could use it in another room and buy a larger rug you will be happier with it. This one is going to irritate you every time the back legs come off the edge.

3 feet provides enough space to circulate around the table. You don't really have much room for additional furniture.

Not every home has dining rooms that are large enough for sideboards and serving pieces.

If I were a wine connoisseur and wanted to display my wines I would look for a really great way to display the wines on my walls and use that as my art.

Thank you for your advice

Design Fan showed two options, and I believe the second one pictured is from Room and Board; they have really nice looking integrated units, all about 25 inches deep. Someone better at space planning will be able to mock that up with your existing table, or tell you if it's a no-go. You could take a look at their site to get an idea...

my link is not linking!

https://www.roomandboard.com/catalog/storage-and-entryway/bar-cabinets-and-carts/amherst-storage-cabinets-with-refrigerator