Leaving the EU will open up a whole range of challenges and opportunities, some of which may see young people ending up slightly less screwed.
If you are referring to housing, the main problem was that there was not enough housing being built (last year ~140 000 houses got built, in 1971 ~400 000 got built), not that there were too many people.
Yes, it's a supply side problem. The housing stock is virtually fixed.
Being a member of the EU undoubtedly pushes demand up.