This is addressed in Percy Jackson and the Last Olympian. In short, gods don't have DNA but nor do they have any of the other attributes that you'd normally associate with a living being since they're godly beings animated by a golden fluid called ichor which has magical properties.
That rule had come up a lot when Silena and Beckendorf started dating.
And I know some of you might be thinking: aren’t all demigods related
on the godly side, and doesn’t that make dating gross? But the thing
is, the godly side of your family doesn’t count genetically speaking,
since gods don’t have DNA. A demigod would never think about dating
someone who had the same godly parent. Like two kids from Athena
cabin? No way. But a daughter of Aphrodite and a son of Hephaestus?
They’re not related. So it’s no problem.
Ichor features quite heavily in the original greek mythos and is a 'blest' (blessed) substance which confers immortality.
Blood follow'd, but immortal; ichor pure,
Such as the blest inhabitants of heav'n
May bleed, nectareous; for the Gods eat not
Man's food, nor slake as he with sable wine
Their thirst, thence bloodless and from death exempt.
Iliad V. 339–342 - Per Wikipedia
This substance is capable of animating even metal (like the protector Giant Talos or the Khalkotauroi)
Father Zeus, surely great wonder rises in my mind, seeing that dire destruction meets us not from disease and wounds alone, but lo! even from afar, may be, it tortures us! So Talos, for all his frame of bronze, yielded the victory to the might of Medea the sorceress. And as he was heaving massy rocks to stay them from reaching the haven, he grazed his ankle on a pointed crag; and the ichor gushed forth like melted lead; and not long thereafter did he stand towering on the jutting cliff.
Argonautica Book 4: The Bronze Giant Talos
As to how this works in humans, the simple (and sadly not very scientific) answer is that the godly component of their composition simply takes care of any problems that would normally be associated with missing chromosomes, the whole thing being hand-waved away by the same magic that allows the gods to exist.