I am hooking up a hot tub, which is 4 wire 240+120 volt, using an old ac line which is 3 wire 240 volt and adding a GFCI breaker as required by the state.
How do I wire that together?
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Your air conditioning line is fine - it is modern with a ground wire. You absolutely need a ground wire for a hot tub, no option.
First I would contact the hot tub manufacturer and ask them why on God's green earth this thing is 4-wire. (requiring both 120V and 240V). That is downright unnecessary. You should certainly voice your displeasure for burdening you with this expensive requirement. Maybe it's not actually a requirement (lookin' at you, Tesla). Or maybe they wanted to save $3 in manufacturing costs by using a 120V gadget instead of 240V. If they sell in the Pacific Rim, particularly the Philippines, they may have an option or kit to support 240V-only. Otherwise I would send the thing back. Inexcusable.
Second I would question whether the circuit can deliver enough power. Most air conditioning circuits are 30A and are wired with 10 AWG wire. A hot tub might be 30A but likely it requires more. Make sure.
Hot tub wiring mistakes are serious business. Aside from obviously killing everyone in the tub or far away, it can kill rescuers too. Electrocutions in water need only stun; drowning does the rest.
If they are laboring to "save the sale", their salesman or tech may tell you to attach the neutral wire to ground. Wrong, wrong, wrong! That is bootlegging, and is just the kind of stupid wiring mistake that kills. Well, deliberately doing it knowing it's unsafe isn't a mistake.
It's legal to retrofit grounds, but not neutrals. You would need to make a new cable run from the box if you can't find a way around the requirement. Then you will be able to choose whatever you need for other uses, e.g. also running the air conditioner.
Most "hot tub" GFCI boxes are 60-amp, and in fact mini subpanels, and some provide a couple of extra slots for additional breakers. Or you could use a real subpanel. So think about what else you'd like to power in that area, and provision the power for all of it.
One option is to use a large transformer to derive the 120+240V with neutral, from the 240V-only supply. It's expensive, but adds a layer of protection by electrically isolating the tub from the supply.
Another is to "hack" the tub (on manufacturer advice) to install a tiny 120V transformer to supply only the 120V for whatever inside the tub needs it. This is exactly how American manufacturers sell dryers in the Philippines, which as a former US territory is slowly converting from our 120/240 to Euro style 230V.
If the hot tub is a 4 wire hookup it apparently needs a neutral for the controls to operate at 120 volts while the heater uses 240 volts.
So the short answer is: You don't hook it up to that wire.
You will need to run another wire if it is in conduit. White in color, the same size as the hot legs.
If it is a cable, then you will need to replace the cable unless the ground wire of the cable is insulated. Is the ground wire insulated?