In Horizon Forbidden West, the Eye of the Earth main quest begins as soon as you find a way up and kill the Grimhorn in the The Dying Lands quest. You begin by continuing further into the facility discovered in the Sacred Cave, but the security system almost immediately detects you, and puts the whole facility in lockdown. So you scout ahead in search of a way to lift the lockdown.
You can actually find the door to The Base secret room while you’re searching for MINERVA. After you pull open a hatch and go through, there’s an open hatch to the right. The Base secret room is through that hatch, but you can’t open it yet, even with the right code. You have to talk to GAIA, ask her some questions, earn the Established the Base trophy, and choose your Subordinate Function quest first, then go back to the locked door.
The Eye of the Earth base door code is 9626118. So, interact with the console next to the locked door and enter that code to unlock it. On the other side of the door, you’ll find five Datapoints: Log: Kelani Jenson, Log: Fayna Leon, Log: Emma Irving, Log: David Makuto, and Log: Ethan Reed.