I didn't answer because I couldn't find any online explanation about "s'emmerder". The fact is we also use "emmerder" to mean to bore/bother/annoy/piss someone off -the latter being the closest as "emmerder" would literally translate as "en-shit" = to pour shit on/over somebody. The reflexive verb is just about oneself.

About what the ditty is alluding to, I see it more as a medical students song (chanson de carabins) -Cf. this article in French about medical students, dissection of corpses etc. "1000 chants" 3, J.E. Berthier, Presses d'Île de France, 1979, has it as a students ditty. (Canon à trois voix sur un refrain d'étudiants)