The translators in the 1500s and 1600s, like those who translated the KJV, were drilled in Greek, Latin and Hebrew for ten hours each day at university by tutors and read every day from a wide library of original Greek and Latin sources - Plato, Aristotle, the Fathers, Thucydides, etc.
Men like Lancelot Andrewes had been reading texts in classical languages for hours each day for thirty years.
Nowadays most people who read the NT in Greek know only NT Greek and have never spent any time reading texts other than the NT in any depth.