Am I the only one who calls bullshit on this? Let's say Elon pulls off going to Mars and establishing a permanent colony within the next 50 years. What plausible doomsday scenarios are there where everyone on Earth dies, but the people on Mars survive and are completely self-sustainable and able to grow their population exponentially?

I mean, even if 99.99% of all humans on Earth are killed, there will be several orders of magnitude more people left on Earth than in the colony on Mars, and they will have a lot more resources to play with.

It's seems pretty unlikely now. But what about once the Mar's colony has been established and growing for 300 years+ ?

What makes you so sure that within 300 years, the Mars colony won't become an existential threat with interplanetary nuclear weapons pointed our way? They may even take to calling themselves "Martians" as generations live and die on Mars with little connection to Earth. Like any civilization they'll have their own language, culture, and values.

The British colonized America in 1607 and 170 years later those colonists were calling themselves "Americans" and waged war for independence. What's it going to be like when that happens on an interplanetary scale?

That's surely a risk. But even in a worst case scenario the chances of a humanity ending scenario are certainly reduced by having more satellites of human civilization.

Could become super handy if aggressive aliens happen to arrive. ;-)

Self-sustainability is definitely more than 50 years away (barring a major technological revolution.)

Once sustainability is achieved, exponential growth seems implicit (even if the base of that exponential growth is close to 1.)

Full, long term Sustainability is probably centuries away (at a bare minimum and assuming some very significant technological advancement.)

Ironically, the very technology (manufacturing, sealed stable ecological design, etc) that would allow a fully self-sufficient mars colony would also significantly reduce that set of possible civilization ending catastrophes on earth.

There are other planets that might be more hospitable for sustainability. It will require much more advanced technology, but even Venus colony might be promising in relatively short run. Not to speak about many icy words around the solar system. If we solve the nuclear fusion, then any icy world with plenty of water would become habitable.