From the control room outside the shelter, you can activate the terminal and use it to unlock the door of the actual shelter. There’s a crate that may contain explosives in the control room. You can explore the shelter deeper.
Continue downstairs and follow the path until you encounter a security door, guarded by a laser auto-turret. There’s a novice terminal on the table which you can disable the turret.
Continue forth and you’ll see an elevator. Once the elevator stopped, you’ll find the room and corridors filled with volatile gas. You can throw a grenade as far as you can then quickly press the elevator to clear the gas and avoid getting burned. Continue forth and deal with the two laser turrets. You’ll find an expert terminal along the way, with a security door and a Novice tool box.
From there, continue downstairs to find the kitchen. Behind it is a storage room. Open the door there and find the Boston Mayoral Shelter Safe Key inside a suitcase in the corner and a Novice locked crate on the shelf. Leave the room and continue downstairs.
There’s an Advanced locked door beside the bedroom. You can find an Astoundingly Awesome tales magazine on the table beside the bed, in the same room with Master Safe. There’s also a steamer trunk beside the safe that contains ammo and some weapons.
Continue along the corridor and you’ll find two rooms. One of them doesn’t have any loot while the other has a Master terminal and a security door with some loot behind it. Before the stairs there’s a Novice large toolbox. Once done, continue downstairs.
Here’s you’ll find a basketball court. Enter the restricted room to find weapon and armor workbenches inside, as well as a fusion core. Entering this room will trigger the synths above so prepare to kill them.
From the basketball court, you’ll find a tunnel leading to a deadend. There’s a deathclaw waiting at the end. Defeat it and collect the Fat-Man and get the Fusion Cores from the Novice ammo box. After looting this, you can now backtrack to the basketball court and go upstairs to the right. There should be a button that will allow you to go through the tunnel.
After reaching the intersection in the tunnel with two skeletons, you can turn to the right to find a monkey alarm and a bottlecap mine. Just follow the tunnel until it leads you back to the elevator. As you make your way to the exit, more synths will attack you, including a Legendary Synth to be prepared to clear them out.