Many philosophers, including 18th century philosopher David Hume and contemporary writers such as Sharon Street and Jesse Prinz, reject objectivism. But there are many different ways to be a non-objectivist.
According to cultural relativism, there are facts about right and wrong but these facts depend on what the members of a society think. For instance, if most people in a society accept that lying is wrong, then lying is wrong in that society.
One problem for this view is that it seems possible to coherently reject the values of one’s society. For instance, suppose that, after learning about factory farming practices, you come to wonder whether it’s wrong to eat meat. It seems that you could coherently do this even while knowing that most people in your society think it’s okay to eat meat. That suggests that to ask whether it’s wrong to eat meat is not just to ask about what most people in your society think.