My favorites have been since I was very young, so have nothing to do with woodturning.
The first is the Grand Rapids Public Museum, in Grand Rapids, Michigan. When I was a kid, I used to go often -- it was free. My favorite part of the entire museum was an entire wing that was called Gaslight Village. They recreated a city street from the turn of the century, lit by gaslight fixtures. It had cobblestone streets, and a horse drawn street car with tracks down the middle. The sidewalks were raised wood planks. The shops were all from the turn of the century, with authentic fixtures, product and mannequins dressed in period attire. I recall a barber shop, an apothecary/drugstore, an old firehouse with horse drawn fire engine, a blacksmith and horse stable. Once a year they opened these shops so you could walk into them, and they were staffed by museum personnel dressed in period costumes. I loved going into that wing!
My other favorite is the Henry Ford Museum and Dearborn Village, also in Michigan. Both these are similar, historically to Gaslight Village, so it's no surprise that I fell in love with these when I first attended them as a teen.
I haven't been back to those museums in several decades, and now live in New England. I still rate the museums I visited in Michigan higher than those I've been to in Massachusetts and New Hampshire. Maybe it's me???😱
I suppose one could find a woodturning connection; I am drawn to pieces that have the appearance of being very old -- like Molly Winton's pyrographed hollow forms with primitive horses, anything created by Clay Foster or Dennis Eliot, the recent pieces by Bill Neddow, and several others whose names escape me before my second cup of coffee this morning.
I haven't been to the Smithsonian in DC yet, but have it on my short list of things to do.