It wasn't even that long ago that these existed. On my first few trips to Japan in 2000/2001 or so, they were still running. I believe it was the retirement of the 100 series Hikari trainsets that ended it - the 100 series kept running for a while after that, but they shortened the trains and removed the dining cars. I don't think the 300 and above series ever had dining cars; IIRC it was just the 0 and 100 series.

The photo above is the 100 series dining car, which is a bi-level car with the dining room using the full width of the car. The 0 series had a single level car with a 3/4 width dining room (2+1 seating) and a small passageway next to it. If I remember right, there was a glass partition, so the people walking by could gawk at the people eating. (Really it was so people on the single seat side could see out the window.)

I'm pretty sure that the last dining car service on any scheduled train was on the Cassiopeia, which I only remember because I rode it and they told us. That was a pretty great experience, especially because I got to do it for two meals when a lot of people on the train didn't even get in for one. (Which I felt bad about, but I mean... get in line earlier!)