Sorry to hear this. Hope that it gets better.

In any professional environment, your boss should not disclose such information with others.
It is really up to you how much you say. The 'bare minimum' should be enough: you are going through some personal issues and you may need time off some time off here and there. Whatever you are comfortable with.

I assume there is not an HR department? HR is usually where you'd say a little more. Correct me if I am wrong, but I believe it is confidential.
Otherwise hearing that it is a small organisation makes me hope that it is at least somewhat closely knit and you know each other and help each other. (compared to a massive corporation)

I hope that this helps.