[unixfox]

Did they carry a battery a few miles down the road, charge it and then carry back a charged battery to give the car enough electricity to get to the next charging station?

Tesla batteries are made up of thousands of cell batteries. Something like 7100 cells. The bottom panel of the car is filled with these cells to comprise the "battery"

It's not like you can just carry a spare with you.

[Cvengr]

Sure you can.
Mercedes Benz AA Class
https://www.youtube.com/watch?v=5sPD5D0EjaI