You misunderstand, I suggest creating a self reliant population actively living below ground (or at the bottom of the ocean). They have to be able to survive indefinitely without help from the surface, but so would a martian colony to weather an ELE.

Do these populations currently exist? Or are you suggesting we build them?

Have you looked into or thought about "breakaway civilization"?

I doubt such populations currently exist, though secrecy would be a requirement to defend against active attacks, so I suppose anything is possible. I'm also not necessarily suggesting we do this, but for those interested in colonizing other planets solely as insurance against major life impacting events on Earth it seems like this would be a more feasible near-term goal to achieve. It would also be an excellent test of the systems we'd need to deploy in the even more remote and hostile environments of other planets.

From my brief research, "breakaway civilization" is a keyword mostly associated with some extreme conspiracy theories concerning the Nazis and the inventions of Nikola Tesla?

I would say some nuclear submarines qualify.

I love subs - they are awe-inspiring, amazing feats of engineering and terrifying weapons.

I highly recommend watching various documentaries on them. I think I have consumed every sub dock I could find online.

https://www.youtube.com/watch?v=wHIS1I9tv78 is a great one. Also, note that the engineering needed to make the Global Explorer was new and significant. I think that with the making of that ship, the CIA gained a lot of know-how for sea-ops, which likely played at least a minor factor into their ability to make requirements/requests into the design of the Jimmy Carter sub, which is largely believed to be the sub they use to splice undersea cables/or cut them...

https://en.wikipedia.org/wiki/Project_Azorian

The sad thing about the end of the cold war is that the governments are spending less resources on establishing such hidden underground societies

What kind of event is in your mind that gives people less than, say, an hour's notice?

I am not an expert in ELEs -- so I do not know... but I suspect if we are using that term, then its an event large enough to screw with the stability of all life populations on earth, thus I state that I doubt that people can evac in an hour - have you ever seen a city evac for a hurricane? let alone for an ELE? Yeah, they're all gunna die.

Full scale nuclear war maybe? Much of the population would only have a few minutes notice at best.

Crazy as it sounds, full scale nuclear war is not an extinction level event. Catastrophic, no doubt, and might reduce the human population by a couple orders of magnitude -- but unlikely to kill off all the humans. There just aren't that many bombs (~17,000 is a high estimate) and the earth is really really big. "Nuclear winter" appears to be overdramatized in fiction.

There's quite a lot of semiserious discussion of exactly this question on the internet; this is just a rough paraphrase of the consensus. It can make for a fun couple hours googling.

How many does it have to kill to be an extinction level event? By the time we deal with the radiation sickness and the lack of civilization would we have enough people left for a breeding population?