There is a very good reason why Earth is, and will continue to, sustain a great diversity of life and the rest of space is sterile.
Is his plan realistic? Hard to tell. It's far beyond anything we have ever done, and financing such a plan is going to be challenging to say the least. And technologically we are also decades out. So I guess he better hurry up. But that's a different goal post that the one I set, a plan doesn't have to be easy or have probable success in order to be better than the other bunker plans.
Also, we haven't even been to Mars, unless SpaceX has some single-state to orbit spaceplane plus some fancy interplanetary engines in the works I'm not that optimistic. And who is going to finance this?
Honestly building a Fallout vault to wait the worst out seems easier.
On the other hand Mars hasn't been stripped of easily accessible resources in the same way Earth has been over the last couple thousand years. As an obvious example, Mars's surface is covered in so much iron ore that it dominates the color of the landscape. Iron isn't worth enough per pound to bring it back to earth, but if they find deposits of more valuable materials there might well be a mining opportunity that pays for the whole endeavour. Nothing brings in money, people and resources like a gold rush.
And honestly, I don't think Musk is planning to do it, my opinion it's 60% marketing strategy 20% wishing for government funding for a small science mission 20% wishing the Mars colony happens somehow.