If you were on Mars, for example, you would not want to have to contact Earth every single time you wanted to load a web page. And so you would contact Earth the first time to load it. And then it would be saved locally.
Don't ISPs already do something like this to save on bandwidth on their side? Just saving local copies of commonly accessed files.
At least I remember hearing about that a decade ago, I wonder if that can still happen now that there's basically https everywhere.
But at any rate, I believe there are at least well established methods for that.
I mean, yeah, you could do that.
I'm not sure if it would be better than a secret underground base... But you could do it.
With an underground base you could even have the one connection to it be a hard-line, not wireless. You could construct it with a smaller crew, easier to keep under wraps. And I expect that would still be less than 1/100th the price of building it on the moon.
Anyway, I do think the ultimate off site data storage location is a pretty entertaining idea, i'd bet it could make sense for some things, I just can't imagine what.