DEE WHY, Australia—In Australian backyards, manicured lawns, flower beds and trampolines are making way for the latest emblem of the country’s booming housing market: granny flats.

These tiny dwellings, also known as “mother-in-law apartments” or “in-law suites,” are the latest craze in a housing boom that started in Sydney and has now spread to other parts of the country, putting homeownership out of reach of many Australians.

In...