A minimum wage should not necessarily afford you a median home, that's why it's called a minimum. But for a functional developed nation I argue it should afford you a private room or a very small apartment. Ideally the cost between the two wouldn't be that different, but due to decades of building restrictions the latter does not really exist. This isn't true in Japan for example, where you can find arbitrarily small apartments at correspondingly low prices.

A living wage is for living indefinitely, not just surviving. That should afford more comforts like a reasonable amount of space, a car if needed, and saving for retirement or emergencies.