You have to be the faction leader, and you have to have a second character with you.
Log in as XYZ and go to a construction foreman, log out.
Log back in as ABC. Go to the same foreman. Tag XYZ. Log out, log back in as XYZ. Buy a diy base.
Assuming you did all that already; this is the guy in Junktown? Cave bases aren't worth it, BUT....... buy him a beer. Chat him up a bit. Then you can buy the cave.
Honestly, an abandoned bunker is worth 5 times as much..... even if they cost ten times.