Honest question. Is this even theoretically possible with chemical rockets?
Lets say you're using Saturn V's to lift people, and you only need 300lbs of payload to LEO (seems unrealistically low, but in this scheme I assume you'll be mining some asteroids somewhere for the interplanatary ships and fuel). Each Saturn V can lift ~850 people. Sounds pretty good. Except that at current rates, 250 people are born every minute globally. Granted, that number would probably slow given universal longevity. If it drops to 40% the current rate, you would need to Launch a Saturn V sized rocket more than once every 10 minutes. That seems extremely implausible, even from a pad maintence perspective. If you had 1000 launch sites around the world, you'd still need to light off more than 1 rocket per day at each of them to keep up.
Ok, so you use something better than a Saturn V, and maybe you can lift 1600 people. Numbers still don't work. Ok, so you take drastic measures to cut the birth rate by another 50%. You're still lighting off dozens of huge rockets every day for the foreseable future.
TL;DR; If you want to evacuate the planet, you need to invest in a non-rocket launch system of some kind. And even then the numbers are going to be challenging.