The Hostel was the Runaways first base, that Chase found while reading about a missing mansion that was swallowed by an earthquake.
Living in a Cave
The Destruction of The Hostel
The Hostel was found and destroyed as a result of the Pride getting information by the traitor of the group. The LAPD found the kids and cornered them. The base collapsed in the form of one of Nico's spells, so the kids could escape.