Translation:I do not have a boyfriend.
It can depending on the context and tone. Determining whether someone means their boyfriend or merely 'he' could only be done in the course of conversation, so if Duolingo uses kare it should mean just 'he'. If they expected people to figure out that they meant kare as in boyfriend from a single sentence without any other context then that would be a bit ridiculous.
I think Duo is just a bit too harsh in those cases where it only accepts が。 This sentence in particular with は means "as for a boyfriend, I don't have one", but you would use が if the context was something like "no – I didn't say I was missing a dog. You misheard me. I said
I was missing a boyfriend".
To build off that, could you also say たくさん人はいます?
I'm also in the camp of thinking this should be が not は, but only because I've always/only seen いる/ある used with が. Is this usage of は similiar to using it to mark times, where its not necessary but draws extra attention to the time being used? (十一時は寝ます vs 十一時に寝ます)