I think most people just use storage facilities to store junk they don't want to throw away, which is a different thing, and often unrelated to the size of your home.
Some people just expand to fill all available space. Most of my neighbors park their cars on the street, despite having two-car garages, and 2000 square foot houses. Their garages are full of boxes, to the point where you can see them leaning up against the frosted glass windows. My grandparents bought the house next door to the one they'd already filled with junk, and filled that one with junk too. I don't think it's necessarily the cost of rent or the size of apartments that proliferates self-storage facilities, I think acquiring junk and being afraid to get rid of it is a different (perhaps especially modern) problem that would exist regardless.