It would probably be cheaper to put it in the caldera of an active volcano than on the Moon. You could certainly get a bathysphere full of archival tape to the bottom of the Challenger Deep for less than you could get the same on the Moon.
Probably about as survivable, too.
Starship, if they solve the remaining issues before politics catches up with them, could solve the first problem.
I've seen some interesting ideas for contact-free drilling that might help with the second, but for now they're experimental* — we've got a lot of things in space tech that need R&D spending, which is a great opportunity on a forum like Hacker News, but does mean dreamers like me need to wait.
>Lonestar’s CEO Christopher Stott says it is to protect sensitive data from Earthly hazards.
If storing your data into 2 or 3 datacenters spread across the planet isn't safe enough from disaster, it's not clear that the moon will be any better since after a global disaster that destroys all copies of the data, it's likely that there will be no one left on earth that still needs or wants that data.