As an individual, can I obtain personal data, such as the private address, of a person working at CERN for my very own purposes?

No. CERN’s data protection framework allows CERN to process personal data only when it is required for the proper functioning of the Organization (see also § 3 OC 11).

Sharing of personal data with natural persons for their private purposes does not qualify as such.

This covers also circumstances where you feel having a legitimate interest, such as collecting an outstanding debt of a former colleague who left CERN without leaving his new address.