This depends on the size of your living space, which varies greatly. Is it really that difficult to fill a 900 sq ft apartment, assuming you don't want to put a lot of boxes in the middle of your living room?
I notice that there are a lot of storage rental places, including major chains.
Assuming. If you're a poor person with a place to live, you don't assume; you put the box.
Where I live, we have nothing like the popularity storage rental has in the US so I can't extrapolate, but I'd guess that it's something that lets you declutter your living space if you can afford it, thereby reinforcing the original point.
At least where I am, cheaper apartments tend not to have garages, where many people put all the stuff they are hoarding instead of their car.
As far as storage rental goes, it's generally available month-to-month, and can be useful when moving. Also, it comes in various sizes, from small lockers to garage-sized, so the expense varies. Say €60-€160/month.
Care/upkeep is usually very minimal, especially with "dumb" items. As for storage, this is an inflexible cost. Your apartment has a fixed surface area for which you pay, and on a day-to-day basis, you only manage how dense you pack it with stuff. If you decide to throw away some stuff, you can't just shrink your house down and pay smaller rent.
This is why poor people tend to own a lot of stuff - because it's really hard to fill your living space to capacity with things, which makes stuff ownership effectively free on the margin.