Could faculty members earn a degree while being employed? Is it possible for faculty members to enrol themselves for a degree while being employed? I have seen professors attending classes occasionally, but how about earning a full degree?

I guess this may not be an issue within a university, but what if the professor wants to earn the degree with another university in the same city, for example?
In France, it is possible, in particular in order to get the "Habilitation à diriger des recherches (HDR)", which is a degree one can get after a PhD, and that demonstrates the ability to fully supervise a PhD student. 
It is required in order to apply for a full professorship position. 

In order to get the HDR, the lecturer must write a thesis and defend it in front of a committee, and needs to enroll in a university (and therefore is a student), usually the one where she is working (but I don't think it is required). Note that for the HDR, there is no class or lectures to attend, only the thesis to write.