This past summer while living in my apartment in Perugia
I noticed a very small water heater right there in the
bathroom. Well come to find out it's not really a water
heater tank per se at all. Apparently it heats the water
as you need it. Water runs through a hot copper tube and
heats the water.

The water heater in my
new house for example is huge since I have 5 bathrooms.
Water heater tanks have the potential to blow up. I know
because my neighbor's hot water heater did just blow up
causing a bad fire. The house is only a year old too.

Tankless Water Heaters by Low Energy SystemsCHOOSING THE RIGHT TANKLESS WATER HEATER - NEW! This brief set of questions and answers will help you get a better idea of the tankless water heater that's ...
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Efficient Water HeatingThe water heater should provide enough hot water at the busiest time of the day. ... New energy-efficient gas-fired storage water heaters are a good, ...
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Tankless water heater are quite common in Italy since metane gas is available almost anywhere. Because electricty cost is veery high most italian are switching to tankless heaters -
This heaters are much more efficient than the standard elecric heater.

In the area where I live here in the US propane gas is not availble so I opted for a close loop Solar hot water system. Note that in many states you can get a refund from your state and you can deduct all or a portion from your income tax.
I will add an image soon if anybody wants to see it -

Are these tankless water heaters capable of heating
as much water as you need? As for example a long
long shower of 1/2 and hour or more.

My water heater tank must be about 100 gallons. Its
huge. Would like to have a solar water heater like you.

Yes - Basically there is no reserve of hot water - Water it is heated on demand - As long as you have gas ( I mean metane gas ) there is hot water - It might be limited in quantity per second: If you have 5 people in the house it migh be able to serve only 1 or 2 with enought hot water to take a shower, but it can stay on all day or forever ... as long as you have gas supply.

My girlfriend is coming to visit, and all she ever says (in a threatening tone of voice) is, "There just better be enough hot water for me & my kids!!"
There is, there is. All you need. No more walking out of the shower with conditioner still in your locks...!

Are these tankless water heaters capable of heating
as much water as you need? As for example a long
long shower of 1/2 and hour or more.

My water heater tank must be about 100 gallons. Its
huge. Would like to have a solar water heater like you.

Villa, the ones in Italy are called caldia , and they are great! The ones you will find in the states are quite expensive and almost cost as much to have a plumber install it as it does to buy it. It is more efficient than the tank type but you have to figure out the overall costs. What do you pay for hot water now and after the cost of the new system. You can google search tankless water heaters and there are some good sites that explain some of that.
I never ran out of hot water and I took long showers. You just have to pick the right size for your house/family.
Ciao,Dan

PS Recently I started working on a new website where the Solar heater is explained - I still have to add images - I have a solar hot water heater at home -

paolo, my brother in law lives in Hawaii and he just put in solar hot water and he says they have plenty of hot water and it is REALLY HOT and they have not figured how to use it all because they don't have a way to store the excess.
Dan