To objectively quantify? As in the thread title? Or to more objectively measure, as in the closing sentence?
It’s obviously possible to improve the objectivity. And you don’t necessarily have to deliberately cause pain to do it. You could study categories of injuries and the number scale answers people give in emergency rooms, with millions of data points, and study the statistics. You could opportunistically monitor things that were painful but predictable and necessary, like vaccinations or medical procedures. And of course you could include numerical signals like changes in vital signs.
You could get volunteers to be hurt, for some amount of money, with the bidding process taken as an indicator – how high will they go? They’d lose the chance if they held out too long, like in an auction. I’m not sure how this works ethically, because they only get hurt when they offer to get hurt, and you’d have to have some way to account for the significance of the money to each subject, as billionaires and people in debt crisis are going to have different motivations.
You could create a ranking scale, which is certainly quantitative even if it has no units, and is I think more objective than the smiley frowney spectrum. Would you rather hit your thumb with a hammer, or have a filling done without anesthetic? People can only answer for pairs of experiences they have had.
In a lot of these, the strength of the improvement in objectivity comes from the statistics. After all, musical tastes vary all over the place, and yet there are lots of objective and quantitative measures of the popularities of musicians.