The Royal Diner is, in fact, a real restaurant. And it is located in Washington... Washington, New Jersey, that is (via Yelp).
The diner, which sits on New Jersey Route 31, doesn't much resemble its television counterpart. On "Bones," the character's favorite eatery is located on a bustling street corner in an urban setting, while you can find the real-life Royal Diner perched along a quiet stretch of highway. However, the interior of both restaurants feature classic diner decor and food items. Similarly, both the real and fictional versions of the diner use a crown in their logo.
Since Booth, Brennan, and their friends love the atmosphere at the diner so much, it's only fitting that customers who've visited the real-life location like it just as well. The restaurant currently has a 4.5 rating on TripAdvisor, with one comment describing it as a "cute, cozy, little 1940's style diner."
It's worth noting that even though the series is set in Washington D.C., "Bones" was filmed primarily in Los Angeles, California, with locations in Exposition Park doubling as the Jeffersonian Institute in the pilot episode (via Movie Maps). It makes sense that the show wasn't actually filmed in America's capital, especially since the real Smithsonian Institution welcomes millions of visitors every year (via Smithsonian). And though "Bones" came to a close years ago, it's nice to know that a piece of the story still exists in the real world.