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
So yeah, like Jack says, supply comes in from lower left: red hot, black hot sharing neutral. The upper right is switched-hot to a lamp.
The upper left carries switched-hot to a lamp, but it also carries unswitched hot on black to other outlets downline.
Important fact: Neutrals don't have breakers. The entire concept is that the hot breaker limits the current that is possible on neutral. Neutral wires are the same size as the hots. The shared neutral arrangement is called a multi-wire branch circuit. Breakers 9 and 21 are both returning current on the same neutral wire.
"But wait a minute, Harper. If each hot is rated 15A, and the same neutral is returning current for both of them, doesn't that mean the neutral is flowing twice as much current, up to 30A? That's way too much since that size wire is limited to 15A." Bingo. That is exactly what is happening in your panel.
"So how is that thing ever supposed to work?" It relies on the use of multiple phases, and the simplest is the North American 120/240V scheme, with opposite poles. If the two MWBC phases are chosen to be opposite, then something cool happens. Imagine black amps as + amps and red amps as - amps. If black has "+" 15 amps, and red has "-" 10 amps, how much does white carry? +15-10 = 5 amps. Only 5 amps. In fact you can try any value for black or red 0-15 amps, and neutral never exceeds 15 amps!
However, this requires the black and red wires be on opposite poles. Let's look at how service panels are laid out.
Now if you look at how your breakers are positioned, it's perfectly clear they're on the same pole, because they're two rows apart.
This also totally ignores another rule that applies to Multi-Wire Branch Circuits: The handles of both breakers must be tied together so they are shutoff together. It does not matter if they trip together, but any scheme that ties them will make them trip together most of the time.
So, the breakers need to be rearranged and replaced with either a 2-pole breaker (Siemens), or you must obtain a Siemens/Murray handle-tie. Then, you must rearrange the panel so the 2-pole or tied breakers can be on adjacent rows in the same column.