2. Adam and Eve
According to the creation story recounted in Genesis, repeated in legend after legend and depicted in art throughout history, Adam and Eve – the mother and father of mankind – would have to be considered an odd couple. They were (after Adam gave up a rib) allowed to live in a Paradise never again attained, for the relatively low cost of simply ignoring the fruit of a specific tree. Their failure to comply led to their expulsion from said paradise, and the all the woes of humanity which descended from them. That Adam and Eve could not enjoy happiness in the midst of all required for their happiness (and upkeep) makes them besides being the first couple, the oddest couple of them all.
Of course, for those who do not take the Bible literally, Adam and Eve, like many of the stories included in the Hebrew Bible and the Christian Old Testament, is not strictly historical. But even discounting their consumption of forbidden fruit and unleashing the evils of the world, they would qualify as an odd couple, having according to the story eaten themselves out of house and home.