You can't correctly say "staffs". I'm not sure about #2. I would not write that. #3 is definitely OK. By the way in music there is also the concept of a "staff". It is the five lines where music writers write notes. When we speak of more than one staff, we refer to two or more "staves". But that would NOT work for members of a staff of the type you asked about.

Sentence 3 is correct. The word "Staff" indicates more than one without having to have an "s" on the end of it. You can say "There are 20 staff members in the university administrative department." or "There were 20 staff members in the university administrative department." The word "Staff" refers to a group of people....so again does not need to have an "s" on the end to indicate it being plural. Sentence 2...."There were 20 staff in the university administrative department." is also technically correct but is used more in the UK than the USA. Adding "members" after the word "staff" helps to clarify the meaning of the sentence,

You can use staffs to refer to multiple bodies when each body is comprised of multiple people. If referring to a single body, just use staff for plural. Use staves or staffs to refer to a long pole, except if it's a compounded word like flagstaff. Staves is more for magical items used in fantasy stories like "the hobbit" though.