Both meanings are possible. It depends on the context. I would say it's more probable that it's referring to two people, maybe 60-40, rather than 100% sure. But also it probably doesn't really matter. Both people were both tired and hungry! If the meaning is different, the sentence will make it clear. - they were both tired - he was both tired and hungry

In English, "not only" and "but," "but also" is a formal way of saying both. For example: Barack Obama is not only the president, but he's also a human being. Not only were they tired from the long journey, but (also) hungry too. They mean essentially the same thing. Except that the not only and but also construction is a more formal way of expressing "both."

The "both" in the question refers to the "they" - for example: They both can sing very well. It gives emphasis that the two of them can sing well. Basically the statement means "Both of them are tired and hungry."