No state now prohibits aliens from owning land. Some states still have statutes prohibiting aliens from owning land; however, those statutes are unconstitutional and not enforced.
It is estimated that there are over 12 million aliens living in the US without legal authorization. Among those millions are many who own real properties.
So, the answer to your question is that illegal aliens can own a home if they have to money or can borrow the money to buy the real property.
Without valid social security numbers, many illegal aliens cannot show good credit via the traditional credit bureaux. However, there are organizations that are willing to look at credit history by other methods.
One of the wonderful things about our democratic capitalist society is that anyone can own real property in the USA, regardless of whether they are legally in the USA or not.