Why is electrical energy most commonly stored in relatively expensive metal acid batteries, which have a short life, rather than hydrogen gas accumulated through electrolysis?
Hydrogen is a versatile energy source itself and with low cost technology is a cheap, clean and reliable fuel.
Using hydrogen efficiently is difficult. If you burn it, you're typically back to IC engine efficiencies, i.e. catastrophically low. If you use it in a fuel cell, that usually involves rare metal catalysts, which are expensive and can be vulnerable to poisoning through contamination. By expensive, I mean platinum expensive, or even more exotic metals like rhodium or palladium.
I don't know what a practical lifetime of a fuel cell catalyst is, in terms equivalent to charge cycles of a battery, but it's an important thing to know if you're looking at the economics of energy storage.
This is on top of the difficulties and expense of storing the hydrogen, as well as the energy losses in electrolysis and fuel cell.
So, let's look at the competition.
By "metal acid" batteries I presume you mean lead-acid, with a lifetime of several hundred cycles - (say, 200-500) thus adding the battery price divided by that number of cycles to the energy cost - indeed making it uncompetitive.
For example, a 12V 160Ah battery stores 12*160 Wh, or 1920Wh - call it 1 kWh at 50% depth of discharge. And costs $200, and lasts 200-500 charge cycles. (Using more than 50% of its capacity will kill a lead-acid battery fast).
I pay 12.3p (about $$0.15 ) for that 1kWh but it would cost me $200/(200 to 500) or $0.40 to $1.00 to store it in a lead acid battery.
Despite this, lead acid are still popular by virtue of being initially fairly cheap, proven and reliable within their limitations, and practical to recycle. Thus they are an easy route avoiding the problems involved in any of the higher tech solutions.
For your H2 fuel cell, you need to find out how much in costs, and how many kWh you can convert through it before the catalyst is junk, in addition to the costs of actually storing hydrogen. I don't know that information, but I don't think you'll like the numbers. Why? because fuel cells have been around since at least the 1970s, and we aren't all using them already.
And lead-acid isn't the only battery.
Lithium-ion batteries have similar lifetimes, (again, say 200-500 cycles) but their weight makes them popular enough that their disadvantages can be tolerated in applications from phones to cars.
However, LiFePo4 cells can achieve as many thousands of cycles, (say, 2000-5000 cycles) even up to 80% DOD, so their higher cost can be amortised over thousands of charge cycles, and this can bring down the cost of storage, especially if their costs come down with mass production.
So, a LiFePo4 battery made of 4 of these cells (about 1.2kWh) would store about 1kWh (80% DOD) and cost about $$560. Then the cost of storing 1 kWh is $560/(2000 to 5000) or $0.28 to $0.11 which finally overlaps with the range of retail electricity prices. (I can't find lifetime specs for this model, so caveat emptor!)
Lithium Titanium Dioxide (often called LTO) can achieve tens of thousands of cycles, (maybe 20-50000) again at slightly higher cost and slightly lower energy storage density - still well above lead acid - this potentially reduces the cost of battery storage further. It also offers a very high charge efficiency : you get about 98% of the charge back out.
We could store 1 kWh in an array of these cells : 5 of them would make 12V, and 4 arrays would give 120Ah to give 1kWh at well under the rated 80% discharge - at about $$1200. Then, over 20000 to 50000 cycles, we are paying $0.06 to $0.024 to store 1 kWh.
(But there's a problem with this analysis : if we're only getting 1 charge cycle per day, 20000 cycles is over 50 years! There may be other reasons the battery doesn't last that long! So use them for more frequent demands; braking/acceleration cycles in an EV, or running the fridge every hour, etc)
So there are batteries offering longer lived alternatives to lead acid, and that does reduce the price of storage, using what's available today. Given some momentum in either improving performance, or reducing prices, they could reduce the niche for hydrogen further. And there's probably a sweet spot for large capacity batteries as cheap as possible with a lifetime about 10,000 cycles...
EDIT : As a footnote, this story holds out the prospect of batteries lasting for hundreds of thousands of cycles.
One of the reasons why hydrogen isn't more widely used is that is extremely explosive requires great care when bein handled. It also needs to be stored away from naked flames or sparks.
The hydrogen atom is the smallest atom and as a H2 molecule it is still extremely small. Being so small, hydrogen molecules can pass through the walls of their containers which requires the walls of such containers to be thicker than for other gases.
Once hydrogen has been stored in a container the main way to use its energy is to burn it, producing heat. If electricity is need to be generated using the hydrogen the heat of combustion needs to be converted by other means into electricity. By storing electricity from alternative sources in batteries, the energy is already in a more usable form.
Cost per watt-hour of storage. Other means are either more expensive to manufacture, more expensive to maintain, or have shorter total lifetime increasing the amortization costs. The one viable alternative currently is pumped storage power plant, but it doesn't scale nearly as neatly for smaller amounts - it needs to be huge to be economically viable.
Hydrogen may seem like a very cheap option if you look only at the very essentials necessary to produce and utilize it, but it being highly explosive, prone to leak through any microfractures and porous materials nothing else can cross, difficult to store in large amounts for prolonged periods of time, and in general very dangerous in handling, necessitates security features which outweigh cost of the actual essential infrastructure costs by orders of magnitude - making the entire solution more expensive than metal-acid batteries, despite core functionality being cheaper.
