That's likely against all sorts of health and safety regulations . Having said that, there lots of small business in the UK being run from storage units - eBay shops, mail order, and even a karate training IIRC.
Anecdotally, there are definitely several homeless customers of the storage facility I use, but I don't think they live there, per se; if only because there are cameras and a check-in/out security system that would catch them if they were sleeping there regularly. I think they just store whatever possessions they still have and use it as a secure place to change clothes, even sit and have some peace for an hour (it's a conditioned indoor building).
Likely illegal in 1st world countries. Safety reasons, zoning, etc.
Also the storage company might not be too happy with you.