The United Nations University (UNU) is an academic arm of the United Nations.
If I understood well, it is doing only research.
Are there any people who are actually enrolled as "students" there?
The United Nations University does not really work as a normal university, but rather as a research center. The difference is that they recruit students from developing countries as for a period that can go from several months to a year, in order to make them work on concrete research projects, so that they can gain experience (which can help them to apply for a Master of PhD program somewhere else).
For instance, this a job position for students at the UNU-IIST (International Institute for Software Technology). Note that the UNU-IIST also plan to offer a joint PhD program with the University of Pisa. Note also that it seems that each institute of the UNU is to some extent independent, so maybe other institutes provide more academic programs.
So, to answer your questions, technically speaking there are students at the UNU (when I was working at the UNU-IIST, there were something between 10 and 20 students), but they are not "enrolled" by the UNU, it's more like some kind of internship.