And yes, it's possible. Am I objectively good at calculus? I passed the AP test. That's an objective measure.
But it's not that simple, for at least two reasons. First, it's always possible to question the "objective measure". Is the AP test a good test of whether someone is good at calculus? AP says so, but that's not actually any help. College admissions departments think so, too. But grad school admissions people don't think so at all, and neither do math department hiring committees. This shows the problem: There is a measure that can objectively be met or failed, but is it the right measure? How can you tell? Not objectively.
Second, in many fields, there is no analog to the AP test. Is Tolkein a good writer? Is the Mona Lisa a good painting? There is no clearly objective measure. The only measure is that many people think so; more people than think, say, my daughter's watercolor is a good painting. That's subjective, but at least it has the merit of being subjectively shared by many people. And with something like Mona Lisa, it's shared by many people across many years - it's not just today's fad. That's still subjective, but it's at least a subjectivity that has widespread agreement to it.