Re: Now go to the Moon
The Russians had a somewhat effective ABM defence around Moscow deployed in the 1970s.
The Americans looked at nuclear-tipped Surface to Air missiles in the 1960s, designed to shoot down incoming ICBMs. They didn't ever deploy them, and I don't know if that was because they were planning to put them in Canada (and the Canadians objected), or if they just didn't work.
But technology has improved massively since then. The Israeli Iron Dome can shoot down mortar shells in flight, which is something that would have been considered impossible thirty years ago. As well as small homemade rockets, and the bigger military ones that Iran give to Hizbollah and Hamas.
Patriot could shoot down Scuds in the 90s, though not to much effect in defending cities. The standard US fleet SAM, the SM3 can shoot down satellites - I presume only the low-flying ones. And is designed for intercepting missiles, but slower ones than incoming ICBMs of course. But it's good at shooting ones on the way up, which is why Japan and the US deploy them off North Korea.
The joint British/French/Italian Aster missiles seem to have similar capabilities to the SM3, which are the ones deployed on our new Type 45 destroyers. Though they're newer and still developing capabilities, so less well proven.
We almost certainly now have the capability to intercept incoming warheads, as we've got good enough radars and communications as well as fast enough computers. Not that it'll ever be easy. But we still don't have the capability to deploy a shield that could deal with the thousands of warheads Russia has. Or even the smaller numbers the UK, France and China have. But those countries have all got small enough warheads that you can have several per missile, and MIRVs, plus decoys. So you could only shoot down some of them.
However North Korea is still making huge warheads, probably in the order of ten tonnes a piece. You can't even drop that from a normal sized bomber. Let alone get even one on a missile. Once they get them down to one-per-missile size, it's probably decades before they could work their way up to MIRVs. So for now they're probably making a handful of warheads a year, that need to go by sea or by truck to their target. So you're only looking at the capability to shoot down a few missiles, which makes it worth trying to do.