You can start the quest through Reilly via Doctor Barrows in Underworld or by listening to the Ranger emergency signal.
Starting it by Reilly is better since she gives you the password for the team's ammo box code.
To awaken Reilly, you need to have 60+ medicine or pass a speech check with Doctor Barrows. It's better to have Barrows awaken her or else Reilly could awaken with potentially permanently crippled legs.
Don't follow Reilly's directions to Our Lady of Hope Hospital. It's better to follow your Pip-boy's directions from the Museum of History, and once you emerge from the tunnels to the Wasteland, head south and find the side entrance to the hospital.
The side room contains a terminal to activate the hospital turrets that could possible weaken the Super Mutants.
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