The practice of storing goods in a unit of space rented from the owner for the purpose, which allows you to have sole access, is not a 1960's U.S. originated concept; it is a practice, which has existed around the world for centuries, if not millenniums.
The practice of renting car garages for general storage has existed in the U.K. Since the 1940's and 1950's, when the building of rows of garages near housing developments became popular.
What the U.S. industry has done since the 1960's is take the basic concept and develop it, improve it and market it. Thirty years later we have a highly refined quality, flexible self-storage product that is universally acceptable as a valuable utility concept by both consumers and businesses.
Whenever the concept of self-storage is introduced to a new market place, in whatever country or region, it has proven to be embraced by the market and replicated. Markets as far a field as South Africa, Brazil, Venezuela and Japan have shown similar development patterns to the early stages of the larger markets of the US, UK & Australia.
In fact Europe's growth trend is showing similar patterns to that of the U.S. in the 70's with growth potential in the short term (5-10 years) in the UK probably being seven times current supply.