Fire hazard photovoltaic battery: What are the installation options?

Battery separately as its own cabinet or as usual as a home power plant? As nice as the idea is, I find it rather pointless in reality: 1. I would either have to move the entire electrical distribution to the ground floor or spend a fortune on wiring 2. Even as a separate cabinet but especially as a home power plant, it is connected with a lot of cables that no one can disconnect without respiratory protection in case of fire 3. Even if 1. and 2. are done, meaning the storage is on a separate cabinet with wheels and without cables and directly next to the ground-level terrace door – who brings it out?

Then of course the fire brigade would have to roll it out with breathing apparatus. But even they would prefer a burning little box on the lawn or on the street rather than the same little box in the far corner of a dark basement.

as usual as a home power plant

"Usual" is not the home power plant, but a separate battery for the inverter.

you hear very little about photovoltaic battery fires, and if something happens, it's a highly marketable headline

and I didn't even know about LiFePo4 until recently - I'm just a layman

LiFePO4 simply cannot technically catch fire. You can drive a nail into it and nothing happens.

However, the comparison with cell phones, laptops & co is flawed. First, we are not talking about batteries of those dimensions, and second, those are much more common than photovoltaic batteries.

No, that's not flawed at all. A burning power bank itself is not a problem, but very few store it properly, for example on a wooden table, on the Sunday newspaper, in another bag, or something like that. THAT then burns. Oil tanks used to be built in masonry "trays." You could do the same with LiIon batteries. You either let them burn or flood them. There's nothing in between. In the roll-out variant, emergency disconnection of the supply lines will probably be interesting too. Under voltage while it burns toxic next door... Therefore: buy a proper battery and don’t worry so much. There are many much more fire-hazardous objects in the house.

Simply do without the memory. It's not worth it anyway :)

Our technical room is about 10 sqm. It is of course already well equipped with heating, possibly a ventilation system, dryer, washing machine, shelves/cabinet. How big is such a battery storage system and what regulations are there regarding where and how it must be installed? I think I once saw someone simply mounting the battery storage on the wall next to the window. It looked like it was just three large car batteries stacked on top of each other? Could that have been a battery storage system? And if so, can it theoretically be expanded simply by adding another "car battery"?

Google "BYD B-Box" or "Pylontech". They usually look more or less like that. Data sheets with dimensions can be found with 2-3 more clicks. I wouldn't really hang them on the wall, unless floor space is very limited and urgently needed, then the effort might be worth it. In a prefabricated drywall room, it tends not to work anyway because they are very heavy.

I think I once saw someone simply mount the battery storage on the wall next to the window. It looked like it was just three large car batteries stacked on top of each other? Could that have been a battery storage? And if so, can you theoretically just expand it by another "car battery"?

That was probably me :-) It's a Sungrow SBR096 storage with 3 modules of 3.2 kWh each. It can be upgraded up to 8 modules. One module weighs about 33 kg.
The "three-tower" has b625mm, h545mm, t330mm and weighs about 100 kg together with the base and lid.