What other ones do you think will be hardest to solve?
I look forward to our subterranean future. I think the combination of vehicle automation and tunneling technology will allow us to build massive underground systems. If things get too rough on land, we can go below the surface.
Why go to Mars to encounter an atmosphere hostile to life when we can all collectively experience right here at home?
Storms washed out your fiber optic cables? Starlink. No place to hide from the hostile surface? Boring Co. Need to remediate the atmosphere? Solar panels / Powersats -- I believe he's being reserved about Powersats. Need to move things around without easy access to transport? Electric vehicles. Need energy storage through the night time because the baseload power stations have all fallen into disrepair? Grid storage. Building your Gigafactory in a desert? Future-proofing.
The most effective answer is distance. While unlucky mobs can walk to your solar panels in the desert after civilization breaks down, how are they going to get to Mars? Get a good trampoline to jump on?
On some Mars colony with the foreseeable level of technology for the next 100-200 years you're even more prone to death and to worse climate than whatever might happen here on earth.
Beyond 200 years, Elon Musk could not care less about...
I believe that he wants to go down in history. And he wants there to be a history for him to go down in. True, he won't be there for that. But the thought of what it might be is a positive motivation for him.
Second, we don't have to worry about protecting acres of land-based solar panels from damage if we're beaming microwaves from orbit. The receiving arrays would be smaller, more defensible, and cheaper to maintain. The powersat could be weaponized to defend its own receiving array.
Third, regarding distance, it's really hard to dig more than a few meters underground without heavy equipment.
Fourth, in a total collapse scenario -- no food, air becomes unbreathable due to methane pollution from melting permafrost --, the people remaining on the surface will die off rapidly. The death squads may last longer, but aren't going to be able to present a significant risk to a deep underground military base protected by the members of the former government with a powersat on overwatch.
After the chaos of the collapse, these locations could quickly begin the effort to repair the atmosphere using all of those terraforming approaches that we would not tolerate being deploying today: ocean iron seeding, orbital shades, bioengineering.
Then the hardened survivors could return to a generally functioning planet, and then maybe go to Mars with their newfound skills.
EDIT: This might seem preposterous, but it's far more viable than shipping one million people to Mars and building up a manufacturing base under time pressure capable of keeping everyone alive and fed while also reproducing a supply chain capable of developing integrated circuits on a world with a different gravitational constant.
Also most collapses to worry about are not "the Earth just became immediately uninhabitable" but instead "there is a complete collapse in civil order". Think the power grid goes down like it did in 2003, but on a wider scale and doesn't come back up. As we go from inconvenience to mass starvation, what happens?
If you hide away, the survivors aren't just going to conveniently die off. They are going to remain, and WILL have a grudge.
If we had room to speculate on the wide variety of disasters that may befall the Earth within the next, say, 200 years, then perhaps hiding underground isn't the optimal best solution.
Aside: I'm sure you'll agree that it'll be easier for Elon to keep his family alive if he's adequately defended. Regardless, he might not have a choice. The third amendment was not always a thing.
As an example of the difference, consider "widespread crop failures followed by food riots causing a general breakdown of transportation networks". The social order is destroyed. But the air is still breathable. And where you live the local crops might even have been fine until pillaged by hungry people.
On a side note, the 3rd amendment has only come up in one legal case, and it was decided in that one that the violation of the 3rd amendment happened, but the government had a qualified immunity because a reasonable person would not have known of the rule. See https://en.wikipedia.org/wiki/Engblom_v._Carey for the bizarre case.
We also use other limited resources in unsustainable ways even if it won't deplete in the next 50 years it is obviously inevitable. I hear about usable sand for construction being depleted at a rate faster than it is replenished.