"I am here without a family" is bad English and would hardly ever be used by a native English speaker. "I am here without family" is much more common, but Duolingo rejected it. This should be changed.
There are multiple reports on this, and as a native AmE speaker, I agree with you. "I am here without family" is now an accepted translation.
So, I guess "rodiny" is the genitive form of "rodina", isn't it? Since other slavic languages use genitive after "bez".