How come every single Cash Run I've been to (about 12 so far), everyone says "Not Available", what do I need to do to make it available?
In addition to not being in a mission, you might have to put away your phone (end profiler by pressing Y) to start cash runs. I had the same problem and this solved it for me. This seems to be a glitch, because once I took my phone out again, it was still available, and it doesn't really make any sense. Still, I'm not the only one experiencing this, and the "not available" message disappeared after putting away the phone.
Also, Cash Runs are dependent on the time of day. Some are only available at night or daytime. Unfortunately, I don't know which ones exactly are affected by this.
Pause the game and press "Quit Mission" and try again. Side missions (i.e Cash Run) should be available now.
Apparently (thanks @Dominic De Coco for this info), the profiler can interfere with side missions as well, so if you still can't access the mission, turn off the profiler (put your phone away).