Concrete Ice Storage

Follow the Building Regulations applicable at that location. Where is the local Planning Office and what advice can it offer?

If you can't get access to foam, acouple of layers of this should work.

You will need to insulate the floor, walls and roof and include a vented air space beneath the roof to reduce solar heating.

Use low density closed cell foam. Styrofoam is very good, but local building regs might not like it for fire reasons. Since ice melts, unless you have active cooling, the floor must be a water proof plate, with a slope to drains. You do not want the foam to get wet, as water can gradually penetrate and reduce the insulation value. You need an external concrete block shell. Built to code, and 10" of foam and floor, walls and ceiling. Have no construction going through the foam. On the inside, panel it with a water-proof layer of material. In a cold climate, you will need to deal with snow loads. I assume you are going to load lake ice in winter to fill the ice hut, and use it over the summer months.

GA

There are very good spray foam insulations on the market with zero VOC vapours. You can get r6/inch which is the highest rated approved insulation. I built my house about 3 years ago and used it with no regrets even if slightly more in cost. Vapour barrier is not needed and the housing frame firms very nicely after it has been installed. Someone mentioned Aerogel and I had not heard of it but googled it. It has lots of nice properties but is fragile which may be a drawback in construction. I am not sure if it can be blown in place, its cost, and availability.

I just read a novel on the invention of ice cream in the 18th century. The book (Empress of Ice Cream) talks about ice houses being built below ground like a root cellar. They barely managed to keep ice in the summer but they did keep it. The ice was from ponds or streams frozen in the winter in England.

sawdust or wood ash make very good insulators.

look into old time ice storage methods. ice houses stored ice year round, with no electricity for cooling

A few sheets of regular insulation wont help very much.

If you can afford it, aerogel would probably work the best.

You are quite wrong, Wood ash and sawdust are not good insulation, they were all they had in the old days. With 12 " of foam you can store ice for a long time as long as you prevent any warm air entering.

That is in fact what tyey do now. They use cheap styrofoam and plastic to keep it dry. A lot cheaper than the pink foam, and a lot better than fiberglass.

You could use earth insulation on the out side- bury the ice house.

The common insulating materials are expanded polystyrene ( thermocole) and poly urethane foam ( PUF). Later one is having half the value of the thermal conductivity of the former. Sandwich panel are available .

Clad the OUT side with at least 100mm expanded polyurethane panels preferably clad with steel painted white,the concrete when down to the temperature you require will act as a battery & even out the minor temp variations also if you get a mechanical breakdown the "battery" will give you a few hours to fix it, as long as you don't leave the door open.

Bazzer

What is the outside temperature where the ice will be stored?

I would look into building with ICFs (insulated concrete forms). They provide an adequate amount of insulation and they can be ordered at most big hardware stores.