This is quite easy as the Church of the new Ark have a unique spire thats unlocked from the beginning called the Temple Spire. This provides high comfort levels so build 1 in each dome. Then just make sure that everybody has access to food and a bed and the comfort levels should easily stay over 70.
The Church has a doubled birthrate perk as well so you'll get plenty of martianborn colonists from the mid game onwards meaning you should hit 250 colonists around sol 75.
The hardest thing about this achievement is the early game as you only have 1 rocket, get no sponser research and recive a low amount for rare metal sales.
I suggest playing with the modifiers prefab colony, fast scanning, fast rockets, increased production & extra applicants. No disasters or events is optional depending on your preference (Both can help or hinder)
None of these options disable achievements.