Based in the Netherlands, they keep all devices in working condition and you can actually use them. Some on request.
What I remember the most was a room exhibiting several older websites on beige-boxes with CRT monitors, running Windows 95 and Netscape 3, and period-appropriate keyboards and mice.
They were set up so that you could click around and browse around, and I even went so far as looking at the source.
What I realized was that NN3 is a pretty capable browser, and so are many which came after them.
Well, I decided to start testing my Web projects with every browser I could get my hands on, and I've discovered that, if you are very careful, you can make sites which work with 25+ years worth of browser software...
And if I'm not on a strict schedule, browser quirks and compatibility are kind of fun!
So I am sort of on the type of mission you're describing. I've set up the appropriate VMs, and whenever I'm working on the computer (pretty much every day haha) is like a visit to a historic browser exhibit. :)
I also like writing in MS Word 97, so this makes it easy to paste the text into my blog afterwards.
Would be great if retro tech museums kept everything in working/demonstrable conditions. That'd be a mission!