First of all, if we are talking about natural disasters, chance is tiny, maybe 0.000001% of a big asteroid hitting the Earth before 2100, big enough to be civilization threatening.
As for extinction causing, not really possible to happen at all, not heard of any credible threat. The solar system is a much quieter place than it was when the big craters on the Moon formed, and there is just about zero chance of an impact that will make the surface of the Earth molten.
Just possibly a global firestorm but some humans would survive that (the dinosaurs didn't have our technology).
So anyway - well if you want to do something about that possibility - personally I think any of those would make a lot more sense than Elon Musk's plan. What conceivable disaster could destroy a colony in the depths of the ocean, or underground? Perhaps a direct strike by a giant meteorite on that very spot?
But then - if a disaster like that worries you so much - just build two such colonies on opposite sides of Earth. They would cost far less than a colony on Mars. Indeed build three or four, or a dozen such colonies - still would cost far less than a single colony of the same size on Mars. And more likely to still be there ten years later (say). And if anything does go wrong with your new colonies - you can get to them to help them out - or rescue them - or whatever, with a trip of a few hours, chances are, not a hugely expensive trip of from six months to over two years depending on relative position of Mars and Earth.
It is not as if Mars was invulnerable to meteorite impacts. Indeed Mars is far more vulnerable than the Earth for two reasons.
It has far less atmosphere (which helps cushion the impacts, most of the smaller meteorites don't hit Earth at all but burn up in the atmosphere).
It is closer to the asteroid belt, which means it gets many more meteorites hitting it than Earth does, of the same size.
So your colonists are far safer from meteorite impacts on the Earth than they are on Mars.
Then you also have to think about - not just the disaster - but what happens after it. If your backup is on Mars then it is in a place with a vacuum for an atmosphere, all oxygen made from ice or other artificial method - or growing algae or whatever, no protection from cosmic radiation, all buildings have to be engineered to hold in ten tons per square meter, can't even leave your habitat without a spacesuit - if your spacesuits are damaged and you can't repair them can't leave it at all - etc etc.
While on Earth even immediately after the disaster you can still breath the air on the surface (even if there is a firestorm in case of enormous meteorite that hits every few tens of millions of years, still after a few hours you have a normal breathable atmosphere again - dark with smoke in the air, but you can breath).
You have cosmic radiation shielding. You have water. You have temperatures in the range pleasant for humans to live in. You don't need a spacesuit. Even in the very worst cases, 0.00001% chance of happening before 2100, chances are you have at least a few plants still growing, most likely whole forests and plains and maybe entire continents that are undamaged or hardly damaged by the impact - and even if not, you can plant seeds and they will grow, soil is still there. Mars of course doesn't have any soil at all in the Earth sense as far as we know. Well has a few streaks of organics here and there as Curiosity found out but hardly soil in the Earth sense.
The obvious thing for any Martian survivors to do is to return to Earth and help rebuild Earth in event of a disaster.
On Earth if you lose technology - you are back to living in caves and houses made of mud and wood and stones. On Mars, you are dead. ~And on Mars things are so much tougher, the technology is harder to maintain - it is likely to be the first place to lose its technology, not the last place.
If you think it helps to live in technologically challenging places - I don't think so myself - but if you think there is some merit in that - well build your colony under the sea. That is technologically challenging enough, you have to maintain your technology to survive. But you have a habitable planet surface just a few hours submarine trip away from your colony.
So why not build your backup colony on Earth in the first place - which is where you would need to go in event of a disaster to rebuild civilization?
Either under the sea, or underground, or floating on the sea, or floating in the sky as cloud nine, or as a desert city - all of those are - some of them quite science fiction - but they are far easier than a Mars colony to build.
I don't see the sense of a backup of humanity on Mars myself. It is not like an external backup of your hard disk in the cloud.
Rather, it is like backing up your hard disk to a USB drive or DVD or similar and then sending it on a spaceship to the Moon. There is no point. The extra difficulty of recovering your data in event of a disaster (not to mention the expense of your solution) far outweighs the advantage of having it in a remote place.
But presumably he is thinking along the lines of the other answers here indeed. Don't think he has really spelt it out in detail, why he thinks this is the way to go.
Or what future disaster it is designed for, and how it would actually help in the event of such a disaster. I don't know if he has worked out any such details, but if so, hasn't said anything about it as far as I know.
I'm ignoring ideas for terraforming Mars here, as they would take 1000 years at the most optimistic, more likely tens or hundreds of thousands of years and that is still very optimistic, it took millions of years on the Earth and Mars is very different from Earth, with much to go wrong.
They are of no value for a short term backup plan for the next few centuries which I think is when we are at our most vulnerable. And if we survive as a technological civilization for 1000 years, I don't think we will need Mars at the end of that period. We will probably know the position of even the smallest asteroids throughout the solar system and who knows what our priorities will be then.
Same is true for space colonies. And colonies on the Moon. I can see value in them, but I don't see them as of any value as a "backup plan" either for the same reason. That even a self sustaining space colony - still - even if they become robust and easy to maintain - almost as easy as living on the Earth - still - Earth will have the edge.
They won't be quite as easy to maintain as e.g. a "Cloud nine" city floating in the Earth's atmosphere or a sea city, and both of those not as easy to maintain as a surface city or country.
So - as I see it anyway - it's only when humans are really well established, stable technologically, then these various technologically challenging space colonies are possible. They won't save us from issues with our civilization, can't see that happening. I.e. they are more of a symptom of success than a solution to it.
So after a disaster you'd always rebuild on the Earth, not in space, and backup colonies easier to build on the earth than anywhere else, if needed. I can't imagine any disaster that would reverse the situation so that it is easier to rebuild civilization in space rather than on the Earth.
May well be other reasons for having settlements in space of course. Just saying this particular one makes no sense to me.
So, for instance, I do see mining in space and returning materials from space as potentially helping with issues on the Earth - and solar power from space as helping. Not guaranteed to work, depends on how it is done. Could make things worse even if e.g. space mining is done in such a way as to out compete the rest of the Earth economy and destroy Earth based industries (e.g. platinum industry). But if done well and responsibly - and if the technological issues work out, I can see that being helpful - both mining and solar power from space.
But that's different - because if those projects work out, they directly benefit the Earth. And if you end up with tens of thousands or more people in space that way - then that does make sense because they are benefiting Earth and have a reason to be in space - and that then is helping with the future of humanity. They aren't there as a backup. They might indeed help if Earth starts to lose its technology, help to maintain it, because they are part of a complete system involving the Earth itself.
Also exploring Mars to find out about origins of life and alternative biochemistry - I see that as potentially of great benefit for life sciences. I don't see exploration and science discovery as pointless at all. But they seem to have no value as a "backup" for humanity with present day and near future technology.
I'm not sure we need a "backup" myself - better to devote our energies to preventing the disasters. We are already doing that with the impact hazards, aim to discover 90% of all potential NEOs of 140 meters diameter or more by 2020.
If we do spot an incoming comet with a few months notice - well we may still be able to divert it, or move people out of the affected zone. If worst comes to it, even with just a few months of warning, we can build underground shelters and seed banks and so on and make sure that as many people as possible survive the impact. I'm not sure there is any great benefit doing more than just make some plans for that right now given the low probability of perhaps 0.000001% of such an event before 2100
But - even if it did happen - we wouldn't launch spaceships to try to evacuate part of the Earth even in that situation - it wouldn't help. We might launch spaceships to try to divert or break up the asteroid if there was any possibility of that. But for survivors, we'd be able to save far more, far more quickly, on the Earth - where everyone also would presumably be engaged in the project of making these shelters and organizing stores of seeds and other forms of life and food - and if necessary oxygen supplies for a firestorm - to survive the disaster. (Something the dinosaurs again couldn't do).
This is separate from my answers about contamination. It's not specific to Mars. I see no value of any kind of space colony, Stanford Toruses, O'Neil cylinders, Moon bases, or anything like that, as a backup colony for Earth, for these reasons. Though there are many other reasons why they might well be worth building.
The Earth may look fragile from space. It is certainly vulnerable to ecological disasters and such like. But it's not going to lose its oceans and atmosphere and its capability to support life in the next few 100 million years. It's not going to become as uninhabitable as Mars. Even we can't mess it up that much.
There is enough time for life to evolve to humans again a second time from the very first multi-cellular lifeforms before Earth is likely to become as uninhabitable for us as Mars is.