Get the money and gun from your office, then go to the market on the opposite side of your garage. Speak to the humanoid robot in the back to buy a battery. (It's the only merchant with wares on its table.)
Go to your garage and use the battery with the car. Then pick up the gas can from one of the shelves in the garage, and bring it to the gas station. Talk to the employee there, and he'll fill you up if you do him a favor.
At this point you'll want to do the "Nothing To See Here" achievement if you haven't already, because you won't be able to do it past this point!
Shoot down and then interact with three different drones in the area. The third drone will have a memory card. Bring the memory card back to the gas station employee, whom will then fill up your gas can.
Go back to your car, and use the gas can on the gas lid on the car's left side, and the achievement will unlock!
This is a story achievement, and can't be missed during a playthrough.