If you get Sydney's help, she'll mention a secret elevator in the rotunda. Use her computer to unlock this elevator and head right down into a safe spot in the sub-basement. Doing this lets you avoid a lot of super mutants on your way into the building.

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If you aren't using Sydney, then you just have to walk around through the office area. You should be able to get into the sub-basement through some stairs and a back door.
You need to go forward toward the loud speaking voice. This will put you in the bunker system with a lot of leveled robots. Stay alert. It's easy to get flanked in this maze. If Sydney gets hurt, you can give her a stimpak through dialogue.
If you have 67 points in Science, you'll be able to automatically fix the utility door ahead of you. If you don't then you'll have to take the path to the left and fight a few more robots to get to the room past it.
Go forward a bit more once you're in this common room. Apparently this room has the central turret generator. I had trouble finding it and didn't see the point. I only saw 4 turrets and they were guarding the Bill of Rights and the Magna Carta. Those are non-essential, so I didn't mind a quick skirmish. If you don't see it then it doesn't matter.
Go down these stairs and through the door. The locked door on the left can be picked to give you access to a caged area. Pick the second lock to gain access to the Bill of Rights. Just blow away the turrets and gather up all the loot and ammo from the safe and the boxes. The locked door on the right should lead to the Magna Carta. It has the same setup.
The rest of this is simple. You just need to move forward through whatever doors you can open. The only big danger is that there are a ton of gas leaks. Be prepared to lose some health from the fireballs or use a melee weapon. After a walk through a few hallways, you should make it to the marked door to the Archive Strongroom. Sydney will point it out too. Save first and then head inside.