During severe cold, DC's Homeland Security & Emergency Management Agency (HSEMA) (a city-level version of FEMA) opens "warming centers" and emergency shelters, typically at schools & recreation centers. They're often managed by the Red Cross.
I've run a few, including one in Maryland during the "Snowmageddon" blizzard.
We had families whose homes had lost power and some homeless. I've found that local authorities and facility managers often don't want homeless in emergency shelters. In a couple of cases, I had to advocate for them or have our Emergency Operations Center (EOC) rep talk to the school system rep or others to intervene, and send orders down for their staff to cooperate. ‾\_(ツ)_/‾ If it was easy, it wouldn't be a disaster! :-)