The Living Computer Museum, funded by the late Paul Allen, does indeed keep their computers running and offer accounts. You can request a login at the following link
The Computer History Museum in Mountain View California has a few old computers running, but is mostly a display museum.
At the Living Computer Museum I logged into a DEC-2060 on a VT100 terminal and amazed my younger friends by navigating around the system and firing up a few ancient text based games. Being of the hacker mindset, I also showed them all the information you could get about logged in users prior to logging in from which one could have used to guess account credentials back in the day...
Explaining and demoing the COMND JSYS was quite fun. The COMND JSYS provided command line parsing and completion, one of the predecessors to t-shell, bash, and other shells with completion. I had a great time revising the past and highly recommend you visit with an old-skool programmer.
For those who are curious you can read about the COMND JSYS on page 164 (chapter 3, pg 52) of the following document
I hadn't realized the connection between the old Boston museum and the one in Mountain View. My recollection was that the collection went to the Boston Museum of Science when the museum in Boston shut down. That's partly true but its warehouse collection had moved to Mountain View earlier and other artifacts later. So the holdings ended up in both places.
Unless I'm mistaken and they both let you do stuff on the computers, which would be cool.