"The old staff"
"Eski" is the opposite of "new," and "Yaşlı" is the opposite of "young;" i.e. the two terms refer to different types of age.
In this case, "eski" can be referring to the staff that no longer work at a certain place (the "old staff"; the people that may have been fired or quit), or the staff which have been at a certain place for a long time (e.g. the veteran staff).
If you were to answer with "Yaşlı personel," then you would be saying something closer to "The elderly staff."
It isn't so much that you can't use "eski" to refer to animate things, than it is you cant use it to refer to people's age.