Electrical – add a service panel in the shed near a generator which powers a cabin

It's interesting, the divergence in design philosophies. On one hand, an all-gas cabin (I live in one) has very minimal electrical needs (we have 30A/120V, and what's amazing is I didn't even know that for 15 years, it just never came up until the lost-neutral incident).

So you have this off-grid home that will never have utility power, yet there's a 4/0 service line (not feeder; no ground) suitable for 225A mains service. And no thought toward power at the shed (short of doubling back with another 12/3 UF cable or something).

It's a misfit for the property... like the contractor didn't know what to do, so just did the standard thing for a utility home. The lack of a ground wire is especially eccentric; you have between $500 and $2000 of wire there; overlooking a $40 ground wire seems silly. Since you say it is conduit (is it all the way or just the stub-ups?) you can just add a #4 ground wire of the same metal, and at least solve that.

I would not remain loyal to any of that, since it is so confused. I would re-think the entire thing from the ground up.

A low-key low-dollar DC system

Given the dollar I see already having been put into this, this may a little lowbrow for you: I mention it because from your stated needs, it would suffice.

Given your modest needs, a low voltage 12/24 VDC battery system may suffice. You run an inverter only for loads which are impossible to run on 12/24 DC. For instance lighting is readily available 12/24V. If you look at TVs, DVRs, internet routers etc., many of those have a wall-wart that actually makes 12 volts DC - you can simply bypass the wall-wart and feed them 12V direct. All power conversions are "vampire loads" even when they are in "standby", so avoiding an unnecessary power conversion is always a good idea. However, a refrigerator is a must-have 120V load -- the people who have been doing off-grid for 40 years have determined that it's more cost-effective to upsize the system a bit, eat the conversion losses, and use any modern, common Energy Star fridge - the appliances are so efficient that $3000 bespoke 12/24V fridges can't beat 'em.

Your well pump obviously needs higher voltage AC power due to the long distances both to the wellhead and down the well. (most well pumps are at the bottom). But that inverter only needs to run when you call for water.

You don't need to use cheapie lead-acid batteries that wear out after 5 years, a lithium pack with good on-board energy management is good for 15-20 years. And for 40 year life, they still make traditional Edison Nickel-Iron batteries (which were for EVs; those were popular in the 1900 era when gas engines were really not ready for prime-time).

In this case, the 4/0 might come into play to bring low voltage solar/wind power from an array or windmill, and/or you could store the batteries at the shed instead of the house.

A midrange system

However, a much more comfortable fit would be one of the midrange off-grid systems - not the Tesla PowerWall but vaguely similar options, possibly assembled from various quality subsystems (Morningstar, Schneider etc). In that case you use a quality inverter that does run 24x7 and is able to support all 120V loads... and just go ahead and upsize the system to support that (smaller because quality) vampire load and conversion losses.

For topping up the large battery pack, you can use a generator, but total life-cycle ownership costs will be better on a solar array, windmill or run-of-creek micro-hydro.

If you run the generator, it can be quite small, since it only needs to meet average demand; the battery covers surge demand. Also, you get to run the generator at a time of your choosing, and it runs near full power for the least time possible, which is optimum for both fuel efficiency and wear/tear. The engine isn't running max RPM no load on the off chance that you open a water faucet.

Speaking of that, if you've got the hills for it, think about a tank-up-high to passively pressurize your water system. Then, you only need the pump to top up the tank, and you can do that at a time of your choosing (e.g. when batteries are full and the solar/wind system is in "dump" mode).

In this scenario, the 4/0 would serve little use, except (if you add a ground) to give you the flexibility to put the gen/battery/inverter system anywhere you please. (but I recommend all in the same place).

By the way... if the cable was direct burial and you needed a ground, just saying, Rigid conduit is an allowable ground path and it only requires 6" cover (12" under vehicle pathways). The shell alone covers the ground, leaving you an empty conduit to fit what you please. It's expensive as all getout, but you can trench it with a garden trowel lol.