Car not starting but the lights turn on?

So I tried to start my car for work and all the electrical components came on, like lights, radio, etc, but would not start or turn over. After a while nothing would come on. I had triple A come, and they jumped it, and it started just fine. I ran it for about half an hour and then cut it off to test if it would start again. At this point nothing was coming on, not even lights. I went and bought a jump starter to see if I could jump it, but it doesn’t jump it just cuts the lights on again. So, at this point I’m not sure if it’s a really dead battery or the alternator.