The photo above is the ubiquitous Gyeongju bread, a sweet red bean pastry with a thin outer layer. It's nothing special, but everywhere you look there are places that make only this item. I'm guessing it's largely that since this is the one food that's specifically associated with the town, and there are so many Korean tourists that the market can bear all those bakeries.
More interesting than the red bean cakes are the UNESCO World Heritage sites of the Bulguksa Temple site and Seokguram Grotto (the whole area gets the World Heritage designation, actually). The grotto contains a spectacular carved stone Buddha (photography not allowed).
Gyeongju is definitely not the culinary capital of Korea. The greater part of its restaurants, it seems, specialize in ssam bap, which is a spread of twenty or so small dishes with rice and various leaves for wrapping them all up in. The small plates are actually the banchan, or side dishes that are served with almost all Korean meals, only instead of about five you get about twenty. A ssam bap meal goes for under $8 per person, but it can get tired pretty quickly, I think.
Earlier that day I had a happier meal of barbecued pork, made from special pigs in overalls.