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.