As-wired, the three wires are Hot, Hot and Neutral. You do not have a ground wire, (unless this run is in metal conduit, in which case the metal conduit is the ground path.)
A 120V receptacle has 3 wires: Hot, Neutral and Ground. Your conceptual problem is you are trying to match 3 wires to 3 wires. And that will never work because those 3 wires have different functions.
First, sort out the Ground question.
Look closely at the conduit carrying the wires. I assume it is conduit because I see 3 loose wires same color. If indeed the conduit is all-metal, and continuous, and joints in good condition, then you have a ground *in the conduit" which should be accessible at the metal box attached to the conduit. It should have a hole tapped 10-32 that you can put a grounding screw into.
You might have an electrician test this believed-to-be-ground path to see if it can really carry a strong current flow. To be actually useful, it must be able to flow enough current to assure the circuit breaker trips.
If you do not have a viable ground path, or if you want "belt and suspenders", you can retrofit a ground: route a copper ground wire from the shed back to the house, via any legal method. If it is indeed in conduit, you might be able to just fish (or even push!) the ground wire through.
The last option is to simply leave it ungrounded. Toss out those NEMA 5 (3-prong) receptacles and buy NEMA 1 (2-prong) receptacles. This will be allowed if the wiring is old enough to be grandfathered (i.e. was installed before grounds became law). If you do this, I strongly recommend you replace the circuit breaker with a GFCI type.
Here's what you can't do: Grab some green tape and mark one of the hot wires as a ground. It is now illegal to remark a black wire (of this size) as a ground. A ground must be green, green/yellow stripe, or bare.
A local grounding rod is no substitute
If this were any more sophisticated, you would need a local grounding system (rods) at the shed. You don't because this is a single (multi-wire) circuit into a single breaker. Here's the point: Do not use a grounding rod as a substitute for a ground wire. You need a ground wire back to the panel so that ground faults will trip the breaker. A ground rod won't do that because dirt is just dirt, it is not a magical superconductor. Dirt conducts electricity about as well as you'd expect, and it won't flow enough current (if it flows any at all) to give you a reliable breaker trip.
So don't even think about using a local grounding rod at the shed as a substitute for running a proper ground. It is a nice add-on, though: what it does is give a very direct path for Mother Earth's electricity - lightning and static electricity - to get back to its source.
Then sort out the Neutral
I see in the service panel where one of the black wires has white tape on it. Similar to green tape, it is now illegal to re-mark a black wire (of this size) to be a neutral. If the installation is old enough it may be grandfathered, and I would take the assumption that it is. In that case, it's OK to use new tape to freshen the mark, as that one is falling off.
It's important the mark be intact on the other end of the wire too. Freshen it there too. It's vital not to get the wires mixed up.
You have Hot-Hot-Neutral. Use it that way.
What you have is called an Edison circuit or a multi-wire branch circuit. This can give you 240V (hot-hot) or 120V (hot-neutral) or 120+240 (hot-neutral-hot) for dryers or subpanels which use both.
If you want a 240V circuit, use the two hot wires. This could go into a NEMA 2 receptacle (or if you have a good ground you can wire, use NEMA 6).
If you want a 120V circuit, use one of the hot wires and the neutral. Do not use the other hot wire. Without a ground use a NEMA 1 receptacle, with ground use a NEMA 5 as in your picture.
If you want a 120+240V circuit, use both hots, and neutral, and ground. If you have ground, use a NEMA 14 receptacle. If you do not have ground, do not install it. 120+240 without ground kills people.
Best Answer
Looks to me like it needs both 120V and 240V.
The 120 is likely for the water pump and controls while the 240 is for the water heater.
Here is a typical wiring diagram for a SPA.
As for what the various current ratings are for on the 240 it is difficult to know without more information. Different heaters for different sized units perhaps, or different heat settings if there are any.