If the words “good” and “bad”, or “better” and “worse” are to mean something at all, we have to directly relate them to the well-being of conscious creatures.
When we say that an action is good or bad we are ultimately making a judgement on the consequences of that action upon one or many sentient beings.
I try to illustrate that there is objectivity in what is good or bad, and hence we ought to act accordingly (that is, there is an obligation for us to act in certain ways). I rely heavily (I would say, fundamentally) in the scenario put forward by Sam Harris, who asks us to imagine “The Worst Possible Misery for Everyone”, that is, a world where every creature capable of suffering, suffers as much as possible for as long as possible and no good comes from that. It is pointless suffering. That would be by definition the worst possible world that could ever exist.
In such an scenario, we imagine then a moral agent* that is given the option to pull a magic lever to one side, removing all suffering from that world; or to the other side, where nothing happens and everything stays the same, leaving all sentient creatures of the world in their unfathomable agony. One could say that there is a strong reason to avoid unnecessary pain and hence one ought to pull the lever so the suffering is brought to an end. But is that really an objective obligation or we just feel like that because we have empathy with the disgraceful inhabitants of that world but it does not really matter to which side we pull?

Feelings of pain and pleasure are subjective, there is no question to that. However, the state of the universe where sentient beings are experiencing something at any given time is an objective datum. If we relate bad and good to the well-being of conscious creatures, where bad equals pain and good equals pleasure, we can then say that, even if feelings themselves are subjective, they are being felt objectively. That is , if someone is in pain, as subjective as it may be, it is an objective fact or truth of the universe that “pain is being felt”. Hence, the moment you say that the words good and bad mean something related to the well-being of conscious creatures, you have to accept that The Worst Possible Misery for Everyone is objectively bad.

In other words, The Worst Possible Misery for Everyone is the ultimate objective badness and, as a such, it ought to be improved by whomever has the chance, since there can be no other reason stronger than avoiding The Worst Possible Misery for Everyone. And that reason to act (i.e., avoiding the pointless agony of all sentient beings) would apply to every moral agent, always, everywhere.
Therefore, we can define objective morality with the set of objective truths that imply that any moral agent ought to act in certain ways when given the option, which would be as objective as “2+2=4”. One example of such truth would be “one ought to try to avoid of The Worst Possible Misery for Everyone”.
We could conceive other intelligent species with different brain structures that do not allow their members to have pleasure and pain, not even slightly, in any form. The fact that they would not see a reason to avoid The Worst Possible Misery for Everyone is irrelevant, because they do not have all the knowledge or capacity to understand it. The moral truth remains the same to them, even if they cannot grasp the meaning of it, in the same way that “2+2=4” remains true for any entity in the universe, whether they understand math or not.
How to figure out which are those moral truths is definitely a hard question (that is what moral philosophy is about) and we may not ever know the answer, but in the simplest case of The Worst Possible Misery for Everyone, the only and objective moral obligation one has, is to try to avoid it.

*By moral agent I mean any entity that knows what subjective experience is, in particular, what pleasure and pain feel like, and also has some ability to produce an intentional change (whatever big it may be) in the order of things.