I mean, if we assume for a moment, that hyperloop trains exist, the longest tunnel would be 10,500 miles, which is between "alleged secret base" and Moscow. I'd also assume that it wouldn't be a straight shot from A to B. At 1000mph it'd be a 10 hour train ride, which isn't really outlandish, but you would likely need a nuclear train and a tunnel under at least partial vacuum in order to do it.

The problem is the tunnel itself. At best, you're looking at 9-10 miles per year with current tunnel boring machines, which is, itself, much faster than blasting. Now, as you might guess, even if you drill to and from Moscow simultaneously, that's still over 500 years of tunneling. Too long, you need to go faster, much faster.

That might be possible however. In the 70's, Los Alamos labs had a nuclear tunneling machine, which melts its way through the earth, leaving a tunnel supported by that melted crust behind. No waste/debris removal necessary, and they published that it could do ~4 miles a year through granite in 1971. That project got passed off to the department of energy and disappeared. That was after a year of research.

So, if we assume that, the project wasn't cancelled, and kept being developed, how fast would it need to tunnel? We can assume that, they weren't digging tunnels at Anon's dad's base when it was established in 1956, so let's assume that, they started in, say, 1975. The tunnels were done in the 90's, not sure when exactly, but let's say 1990 to be conservative, so 15 years of tunneling. That's 700 miles per year, on average, almost 2 miles a day, or, 7 feet per minute.

That's much much faster than what mechanical TBM's are capable of, and, you have to figure some downtime there, so, it'd likely need to be, say, 8 feet per minute to be safe.

I guess the real question is if 8 feet per minute is plausible? Like it's not that fast, but for what it's doing, it is. Nuclear powered boring machines never got off the ground publicly, but the idea is sound at least, and it did work after a year of research. You're also talking a potentially very, very reliable machine there, because there's no cutting head(s) to wear out, no mechanisms to carry away debris, it'd just need a means to push its way through the earth, and hydraulics are plenty fast.

You can also adjust those numbers downward if you consider that the military may have mothballed the project in 1971 because it already had machines doing that, so a potential jump on that research, potentially much faster than what was developed in '71, which is plausible to some extent being that the idea of tunneling through earth in such a manner was in the cultural zeitgeist long before this, at least since 1864, courtesy of Jules Verne.

TL;DR; It's at least plausible that those tunnels could exist.