Mimi Your last sentence is okay from the standpoint that a listener or reader would know what you are trying to say. However "at a place nobody knows" grammatically cannot stand without the modifier, "that" or "which" and I think "which" preferable. We run into the same thing in French. Often I will draft a sentence in French and realize it needed "que' or "lequelle."
I think Deloris' suggestion is okay, and so is Stezza's (unknown place), even though Stezza didn't like it ... but the real problem to me is your demand for "at a place" followed by some description. If it's absolutely required, then Homer was born at a place nobody can identify, or nobody can determine. But as everyone has pointed out, it's a little off. Still awkward, but possibly more common, would be "He was born in an unknown locale." Best of all, in my view, was Stezza's original suggestion "Nobody knows where Homer was born," or even better, "Nobody can say for certain where Homer was born."