The chuck I have is: Jacob's 0-1/2" JT #33
The lathe I have is: G0768
I'm wondering what chuck arbor I need to buy / how to determine that. Related: the chuck has threads in its hole, but I see no arbors with threads, so what's that all about?
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According to the spec for your lathe in the link you provided you need a MT2 arbour. What is a bit confusing is that that model of chuck is supposed to have a tapered fit to the arbour rather than a threaded one. JT#33 is the taper fit to the chuck and MT2 is the taper fit to the tailstock.
Threaded arbours certainly exist but you need to determine what thread your chuck has, that size is probably either 1/2 x 20 or 3/8 x 24. However taper mount chucks are far more common for lathes.
So a JT#33 chuck is what you want but it sounds like that's not actually what you have got.