Before We Leave

For this achievement, you are required to advance through the tech tree until you can research Shelters. This technology comes quite late. Once researched, you can build Shelters. You are required to build a shelter on each island that is inhabited on your planet and hope for a Whale attack on that planet.

Your Peeps will automatically go into hiding so no action is required on your part. However, the planet must not feature a Space Elevator or an Aegis Shield. If it does, you can turn those off. I would also recommend building your Shelter somewhat central and with good access, meaning not constrained by elevator usage.

Once ready, a Whale attack needs to hit a settled island. If it does and no Peeps die, because they made it all into the Shelter in time, you will get the achievement. Be advised that this will cripple the island and will come with some, possibly, heavy infrastructure damage. It may be best to wait till the late game and save, then disable better defenses, to get the achievement without really impacting your playthrough.