The Robot Crab is directly outside the Living Quarters door at the left side of the island. However, to enter the quarters requires a separate key, which is found at the far, far side of the island, in the Research Labs. This requires that you solve more than half of the island quest before you can get the key.
(see solution at related question)
Basically, these are the steps to reach the Living Quarters key :
Go to the Living Quarters and open the door using the Steam Battery as with the previous entrances.
There are no windows that open inward on Steamworks Island. There is a skylight in the City Hall that can be opened with the Multi-Tool. You go out the hatch above Zack's room in the living quarters, to reach the Mech Crank on the vines.
The steam battery is in the trash bin at the right of the Cantina. You need it to open the doors (Sully's, Living Quarters) and lower the Bridge at the Hub.
You don't put it anywhere. Like the keys for Sully's and the Bridge control, you open the door using the Steam battery. You just need the key in order to use the Battery.
You use the same Steam Battery to open the two doors (Sully, Living Quarters) and lower the bridge. But in each case you still need the key. Sully's key is in the Robot Crab, the Bridge Key is in Ziggs' room atop the Hub, and the Living Quarters key is (perversely) in a closet at the Research Lab all the way to the right of the island. Getting to it is more than half of the quest.(see the related questions)
After completing the island, jump down the open shaft at the far right of the elevators. This will take you to the exit to the Living Quarters scene. If you end up on the bottom level, take the "elevator plant" at left and jump across to the right on the next level.
Both of them are left of Main Street (Cantina and Museum). But to open the doors, you need the Steam Battery from the trash at the Cantina, and you need a separate Key for each of the doors, and to lower the large bridge at the Hub. Sully's key is inside the Robot Crab at the Living Quarters. But the Living Quarters key is far across the bridge in the Greenhouse (Production zone). Getting to it is more than half the quest. (see the two related questions)
Both of them are left of Main Street (Cantina and Museum). But to open the doors, you need the Steam Battery from the trash at the Cantina, and you need a separate Key for each of the doors, and to lower the large bridge at the Hub. Sully's key is inside the Robot Crab at the Living Quarters. But the Living Quarters key is far across the bridge in the Greenhouse (Production zone). Getting to it is more than half the quest. (see the two related questions)
The only reason you need to go there is to get the "special number" that Captain Ziggs uses as his combination in his quarters. It is on the desk :the number 516 is translated into "0516" to open the secret closet.
The flying robot drops the robot crab on it
There are three keys that allow the Steam Battery to open locks. The first is Sully's Key, which is hidden in the Robot Crab outside the Living Quarters (far left of the island). The second (Drawbridge Key) is in the secret compartment in Captain Ziggs' room, which can only be accessed when you have Sprocket and the robot crab to cool the lower hatch cover. The third (Living Quarters key) is in a Maintenance Closet inside the Biological Lab (far right of the island) To reach it, you must use the drawbridge key. (see the related questions)
There are three keys that allow the Steam Battery to open locks. The first is Sully's Key, which is hidden in the Robot Crab outside the Living Quarters (far left of the island). The second (Drawbridge Key) is in the secret compartment in Captain Ziggs' room, which can only be accessed when you have Sprocket and the robot crab to cool the lower hatch cover. The third (Living Quarters key) is in a Maintenance Closet inside the Biological Lab (far right of the island) To reach it, you must use the drawbridge key. (see the related questions)
The Steam Generator (to open the doors) is in the trash bin to the right of the Cantina.