Mrknownothing said..
Brains trust.
BA Falcon fun..
Car wouldn't start. Checked battery. Battery is green/red, which is apparently fully charged. Thought I'd try and jump start her anyway. Car starts. Took for a fang along the motor way. Turn car off, car won't start again.
In my non mechanical opinion, I'm thinking alternator. How can I check this? And is this something I could attempt myself?
Sure, you could attempt an alternator change yourself. Your results would depend on your experience and skills.. just like everything

They come out from under the car, and you need to remove a bracket, so its not trivial. The access to the belts is not easy either as you probably need to remove the fan/shrouds as well.
The good news is that the BA doesn't have a smart alternator, so they will be producing the same voltage all the time, unlike the BF and FG where the alternator output is controlled by the ECU.
What you describe seems to suggest the battery though. If it won't start with the battery 'fully charged' but starts with a jump start, that tells you something. If it then won't restart, it is either the alternator or the battery, but you cannot tell without knowing if the battery is good or if the alternator is charging. You would have to charge the battery with a battery charger or from a known good car to then see what the behavior is.
These days I treat batteries as consumables. If it has more than 4 years on it, replace it, they are cheap and easy to swap. If the battery fails to charge, it is then the alternator, but you know you now have a good battery for another 4 years.
Falcons do have a known issue though where powersteering fluid drips down into the alternator and ruins it. I think it gums up the brushes.