Public housing generally isn't the answer if the public housing stock is poor. The tiny environments have a serious impact on Hong Kong society, especially when we're talking about birth rates and distribution of services. Its not like they don't have land to use, its just land policy is locked and land that is available is being exploited by developers and investors.
This editorial (yes I know its SCMP but he's said nothing wrong) best highlights the ongoing struggle between society and the government/developers - as free and wealthy the markets may be, the quality of life is not particularly good.
The point is that adequate housing is extremely, extremely expensive. Even if you work, you can only afford a tiny square about the size of a typical Australian living room. Which costs half a million USD, which is huge for the wages that most Hong Kongers actually earn.
The reason being those prices are due to accepted land market speculation and poor land policy. The 2nd can be solved by changing how land use and development is approached; the first is solved via regulation, namely vacancy taxes and better regulating land valuation.