The ESP/BAS light indicates that there is an issue with your Electronic Stability Program (ESP) and/or your Brake Assist Program (BAS). These safety systems work together, and a fault in either, or both, will cause the trouble light to come on.
The steering angle sensor is one of, if not the most, common causes for the ESP BAS light. You can try turning the steering wheel twice to the right, twice to the left, and finally back to the center to reset the steering angle sensor. Just like with any other major vehicle problem, you’re going to want to get a reading of your diagnostic trouble codes.
Another possible cause of the light is a defective wheel speed sensor. I had this happen on my 2009 SRT. The ESP and BAS systems measure the speed of all four wheels at all times. When the wheel speed is no longer equal, they will apply corrective measures to make sure that you stay in control of the vehicle. These are easy to replace, and have their own diagnostic code.
Lastly, something as simple as the brake switch is known to cause the ESP BAS light to come on. Replacing it is simple and cheap.
I don't know why your car stopped running. Can you start it now? The problem causing the ESP BAS light is not going to be engine related. Driving with the ESP BAS light on is not going to leave you stranded on the side of the road. It is, however, not as safe. It’s like driving a car from the 70s that doesn’t have traction control or ABS. You don’t always need these systems, but when you do they can save your life and keep you from crashing.
I had this happen recently past 2 weeks its a Wheel Sensor about a 30.00 to 40.00 part depending where you get it O'Reilly's, Autozone, Advanced, Pep Boys, etc.,
You need to get it on a Scanner to determine which one of the 4 it is triggering it Left Front, Left Rear, Right Front, Right Rear only a Scanner will pinpoint it.. Mine did not shut car off but did Go on around a corner... All together I paid 90.00 to resolve it. Sensor below in Link. ...This turns on those BAS/ESP light its a common issue.
I just wanted to add to this post just in case someone in the future comes across it. I'll explain the symptons but in the end it was the alternator going bad. I was having similar issues as the OP did. It first started out where I noticed my breaks pulsing weird at a certain pressure. Days later I heard a slight whirring sound and my check engine light came on and when I checked the scanner I got a P0013 Code which was Exhaust Camshaft Position Actuator Circuit Open , Bank 1 which has to do with the Variable Valve Timing Solenoid. The car also was rough to start. After checking online it said those could be symptoms of the solenoid not working properly. The next day the car was running fine and I drove around 60 miles with no problems. The next morning the slight whirring was back and I noticed my engine temp got up to normal operating temp very quickly and then it was slightly higher than normal. I started getting my ABS and EPS lights to come on and the traction control light. They would go on and off and I would get the ding noises and when I would turn the wheel my tach would stop working (2010 with analog guages). Luckily right when I pulled into the driveway the car died and when I opened the hood I noticed the slight burning electronics smell so then I figured it was the alternator. Did a little testing to confirm and ran to the parts store and swapped it out that night and all of those problems went away.
I just wanted to post this just in case someone has these same problems and to check the alternator first before getting some other uneeded repairs done and then the problem still persists. Luckily when I went to the dealership to have the solenoid replaced they were back logged for weeks and couldn't work on the car, that would have been money down the drain.