To obtain a Chaos Gear, you need to find and kill Ruin Sentinels at any level. Upon eliminating these automatons, they have a chance to drop Chaos Gear.

You can get these from the none Humanoid Ruin Sentinels.

Ruin Sentinels are only found on Inzauma, on almost every island. The fastest way to get around is to use the Waypoints, kill, and move on. The image below is where you can find Ruin Sentinels in Inazuma. There are also eight at the tip of Narakami Island.

Comes from a now-inactive ancient Ruin machine. This is the energy source of that biomimetic mechanism.