...or asian, or hispanic, etc. And yes, a man can understand what it is like to be a woman. A straight person can understand what it is like to be gay.
Did I get your attention? Cool.
I've seen this question in the race conversation a lot - everybody has - and I think the answer is clearly "yes". The important part of the question, though, is that race has nothing to do with it.
The real question is "Can one human being understand what it is like to be another human being." Race is just one variable in our experience as human beings. Gender is another. Dis/ability is another, as are age and economic class and culture and religion and anything else you can think of that describes a person. The question is, can we understand each other? The answer is yes - it has to be yes for any of our communication or relationships to be meaningful at all. If the answer is no, then apoia'giajds;aio hgdag;ieha.
Now, if the question is "Can one human being experience the experiences of another human being", the answer is obviously no across the board. We can't live another person's life - we can't be them. But we have more than just experience to help us understand other people.
We have limbic resonance. Your mammalian brain broadcasts how you feel through subtle clues many times per second, thousands of times in a given conversation, entirely outside of your control. Outside of my control, my mammalian brain picks up on these cues. Happiness, sadness, anger, even things like obesity, are actually contagious in this way. Our mammalian brains are always striving to understand and communicate with each other, and this process is not impeded in the slightest by something constructed like race or ethnicity.
We have empathy. We have the capacity (with the exception of sociopaths or perhaps severely autistic persons) to feel what other people are feeling. When we see a sad face, we feel a pang of sadness. When we see an angry face, our pulse rate goes up a few beats or more. Reveling in empathy isn't very helpful, but it is there. We know from the research of people like Charles Darwin all the way to Paul Ekman that emotional facial expressions are universal regardless of culture. If you haven't read Paul Ekman's books, I highly recommend them. He has demonstrated through decades of research and facial analysis that the facial expressions that accompany emotions are universally human and entirely cross-cultural. Culture teaches us different ways of managing those emotional reactions, but the reactions go deeper than culture or ethnicity, all the way to biology. We are 'wired' to communicate emotional experiences and emotional states, whether we are New Guinean stone-age hunter-gatherers or Japanese executives.
We have sympathy. This is the imaginative capacity to understand, from a slight remove, what it is like to be another person. We can find experiences in our lives that are similar to experiences in other lives. We can listen to their stories and imagine ourselves in those stories. We can ask them what it is like to be in their skin and the words they tell us have meaning. We have to be careful that we're not just projecting our own experience and biases onto the other person - but that's what listening is for!
We have imagination. We can imagine, and understand to a degree, what it is to be a big blue alien on Pandora, or an Elf, or a cyborg, or an artificially intelligent robot, or a killer whale, or Jesus, or our parents, or our ancestors. We started developing this capacity when we were toddlers. We can use it to manipulate others or to help them, to build up or to destroy, but we have it. We can walk in each other's shoes just like we can walk on Mars or the bottom of the ocean. More information and broader experience sharpens this capacity, as does wisdom, but we have it almost from the beginning.
What we cannot do is to have the same experiences as another person. We can't mind meld or download their experiences into our brains. So, while I am not black, not a woman, not gay, not hispanic, etc., can I understand what it is like to be those things? It takes some effort and imagination, but yes. Can I have the experiences of those other categories of people? Can I embody them? Can I be just like them and live their lives? Not at all - of course not! No more than they can embody me or have my experiences or live my life.
So, the bad news is, we're just going to have to listen to each other, and actually use our capacities for understanding.