The Outsiders have used several different locations as their headquarters over the years.
Their original headquarters when Batman started the team was a meeting room in the underground location of the Wayne Foundation, which had previouly been used as the Batcave II. They lost this when Batman quit the team. King Gregor later agreed to fund the team and built them a new base in the Pacific Ocean outside of Los Angeles. This was a research facility named Station Markovia run by scientist Helga Jace. Batman would eventually reconcile with the team and build them a base called Batcave West. Major Disaster destroyed Station Markovia while working as an agent of SKULL.
When Arsenal established a new Outsiders, he used an abandoned bomb shelter underneath Brooklyn as their headquarters. This was stocked as a state-of-the-art underground military bunker. Arsenal and Jade blew up the bunker when they decided its security had been compromised too many times. Batman later took control of the team again, and had them operate out of the Batcave. Alfred Pennyworth took over the Outsiders when Batman disappeared, and used the orbiting Bat-Rocket as their mobile base.