The Bias of Bias

From a young age, my greatest pet peeve has been variations of "nobody's perfect", or supposedly more academic formulations such as "nobody's objective" and "everyone's biased". The latter claims are especially common among those who profess scientific inclination, despite being obviously, in their own words, "unfalsifiable": How could someone disprove those claims? What evidence would satisfy them that someone is completely objective or completely unbiased, or indeed, perfect? Yet, the problem is more fundamental than that. Those claims are self-contradictory. "Everything is relative" cannot be true as the statement is self-refuting: If "everything is relative" then the statement "everything is relative" is also relative. Likewise, if "everyone's biased", then how was that statement itself established without bias, and if "nobody's objective", then how was that statement itself established objectively?...