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.
Go ahead and do that
A subpanel downstream from a transfer switch is generally an expected thing to have in a standby power system. 125A is a fine size for such (since you likely aren't going to be putting nearly that much load on the standby system), and a 24 or 30 space unit would be not at all out of place here, either.
As to neutrals? It's the transfer switch's job to deal with any neutral shenanigans that may be going on. Since you are using your generator mainly for backup, I'd pull the neutral bond and put a clearly visible, indelible label on it saying that the bond was pulled and referring to the correct section of the manual to reinstall it if I were in your shoes, by the way -- this means you can have a solidly bonded neutral in your transfer switch, which makes life far easier when it comes to transfer switch selection.
Best Answer
No "checklists"! Hard-built interlocks only.
There is no such thing as "making sure your main breaker and 2-pole breakers are turned off". You must not do generator interlocking via a checklist or procedure.
There are only three legitimate ways to switch from mains to backup power:
The cheapest and simplest way to implement a generator interlock is, if an interlock kit isn't made for your main panel, to get a no-lug subpanel with an inexpensive interlock (Murray or QO) and have two backfeeding breakers back to back interlocked, and then put the loads you want to switch into this panel.
Go ahead and feed both poles, except...
Once you have done this, go for it. Go ahead and split your 120V supply so it feeds both legs of the interlocked 240V "generator" breaker.
... except Multi-Wire Branch Circuits
However, this will be very bad for multi-wire branch circuits (MWBC). These are circuits which share a neutral. The hot wires must be on opposing poles so the neutral only handles differential currents. Putting them on the same pole will mean neutral handles the sum of both hots' currents, and that can overload it.
How do you avert this? Ideally, you put your to-be-gen-powered circuits in a separate subpanel (again, the cheap way to provide that gen interlock)... and simply don't migrate MWBCs to that panel.
Or go on a crusade to eliminate all multi-wire branch circuits,
At the very least, you identify in advance your MWBCs, by going through your panel, finding them, and placing them on 2-pole breakers.
Notably, a "make turning off MWBCs part of your checklist" apparently violates the "no checklists" rule. However the main reason for that reg is so you don't kill linemen, who are innocent outsiders. I am not sure if Code has taken a stance on this entirely local consequence.
What will happen if you power a 240V appliance?
A 240V appliance (water heater, A/C): Nothing. Every conductor in the appliance will be at the same potential, so no electrons will move. This will happen to be 120V from safety ground, but who cares.
A 120/240V appliance (dryer, range): The 120V parts of the appliance will power up (tumbler, timer, clock, oven light). The 240V parts, see above. So for instance your dryer will work, but only on "fluff/no heat".