Who is a ‘retired goat drummer’?
In the old days, one popular pastime was watching bear performances, where the animals could perform various tricks: they'd impersonate a drunken peasant, a swaggering gentleman or a mother-in-law treating her son-in-law. One of the "bear centers" where the animals were trained to then perform at fairs was Sergach in the Nizhny Novgorod province. Read about how the French welcomed a real bear regiment to Russia here.
A special person trained the bear – a ‘medvedchik’ (lit. ‘bear man’). He also had two assistants. One, dressed in a "costume" made of a sack with a goat's head, impersonated a goat that jumped around the bear. The other played the violin or drums. The job wasn't difficult and you just had to try hard to lose it. If this happened, the assistant became a “отставной козы барабанщик” (“otstavnoy kozy barabanshchik” or "retired goat drummer").
Over time, this expression came to be used not only for traveling performers, but for anyone who was doing nothing, scraping by with odd jobs and was otherwise unremarkable. Such a person could also be described as a "zero without a stick".