Without being able to see the cables as they enter the cabinet; or the ability to touch or trace them, here is what I assume is going on.

Definitions:

Grounded (neutral) from the service

A typical single split phase service is made up of 3 wires. Two ungrounded (hot) conductors, and one grounded (neutral) conductor. The ungrounded (hot) conductors will connect to the main service panel through a disconnect (usually a large breaker), while the grounded (neutral) connects to the neutral lug. The neutral lug will be bonded (electrically connected) to the neutral bus bar, and all grounded (neutral) branch circuit conductors will terminate at the neutral bus.

Grounding Electrode Conductor

This conductor is used to connect the grounding electrode (ground rod, etc.), to the grounding bus in the panel. All equipment grounding conductors will be connected to this bus.

Bonding Jumper

The bonding jumper is used to bond (electrically connect), the un-energized metal parts of the panel to the grounding system.

Assumption:

Since it appears that (what I assume is) the grounding electrode conductor terminates at the neutral bus, I'm also assuming that this is the main service disconnect. This leads me to believe that the neutral and grounding buses are bonded (electrically connected). In which case, technically, grounded (neutral) branch circuit conductors can terminate at the grounding bus.

So you have two options:

1. Terminate the grounded (neutral) from the new circuit to the grounding bus.

2. Move the green wire that is terminated on the neutral bus, to the grounding bus. Then terminate the grounded (neutral) from the new circuit, to the freed up slot on the neutral bus.

Additional Information and Code Compliance:

Number of Conductors

Since this is a new circuit, it has to be installed to current code standards.

National Electrical Code 2011

ARTICLE 250 — GROUNDING AND BONDING

VI. Equipment Grounding and Equipment Grounding Conductors

250.140 Frames of Ranges and Clothes Dryers. Frames of electric ranges, wall-mounted ovens, counter-mounted cooking units, clothes dryers, and outlet or junction boxes that are part of the circuit for these appliances shall be connected to the equipment grounding conductor in the manner specified by 250.134 or 250.138.

Which in this case means installing a NEMA 14 receptacle for the dryer, and a proper grounding conductor.

You'll have to follow the dryer manufacturers installation instructions for upgrading to a 4 wire cord. For more information see this answer, and this answer.

Since you've said that you're already using 4 wire cable, you'll simply have to terminate the grounding conductor in the cable to the grounding bus in the service panel. Then connect the other end of the grounding conductor to the grounding terminal in the dryer receptacle.

Size of Conductors

You'll also want to be sure that you're using the proper size breaker and conductors. In the case of a dryer, you'll typically use a 30 ampere breaker and 10 AWG conductors (depending on the length of the run). However, you'll want to check the dryer manufacturers installation instructions to verify this.

If the barn has its own main service

If the power company is supplying an electric meter that powers only this structure, then you're sympatico. This panel is a main service and this is the normal way to wire that.

However since this is the home improvement forum, I assume a home is involved.

If the barn is fed off a main panel in another building

Like the windmill said to the small child: "I'm not a huge fan."

You're in a no-win situation with your grounding.

The principles

Grounding serves at least two purposes: To dissipate ESD and lightning strikes back to the earth, and to return hot-ground fault currents back to source with enough current to trip the breaker (50-100 amps). If the latter doesn't happen, the fault will try to "light up" the grounding system and shock people (0.1 amps kills).

A grounding rod was never even imagined to return fault currents. It just can't. Dirt is an unreliable conductor, which is why they don't use it for wire. Dirt can flow 0.1 amps, but 50 amps is not gonna happen. It's good for grounding ESD and lightning, and for pegging neutral to earth potential to make a non-isolated system.

Second, if the neutral wire breaks, you have a classic "open neutral" problem: the 120V loads are not equal, and they pull neutral toward one pole or the other... the effect of this is to make neutral hot.

Here are your choices, Sophie

One choice is to tie neutral to ground. This will assure that a hot-ground fault trips your local breaker. However, an "open neutral" electrifies all your grounds. Every bit of conduit, all the switch plate cover screws, all equipment chassis, even the subpanel door! The grounding rod will valiantly try to return this current via the dirt, but it can't win.

The other choice is to wire it as an isolated system and intentionally isolate ground from neutral. Except it's not an isolated system, is it? Neutral is pegged to ground back at the main house, and by the power company on the pole. Yes, now an open neutral will not shock you. But hot-ground fault will! It will pull the grounding system back up toward 120V without flowing enough current for a breaker trip... so you're getting shocked again! You can't win.

The right way

It's perfectly safe and legal to retrofit a ground. Hit the hardware store and get some copper (can't be aluminum) and run it back to the house and tie it to the main panel there. It doesn't need to follow the same path, just needs to use a legal wiring method.

I note that #2 ground wire costs nearly a buck a foot and needs to be buried 12". So does Rigid conduit, which can be buried only 6" deep. I also note that the metal shell of rigid conduit is itself a legal grounding path.