Is it feasible to synthesise sound with analog circuitry these days?
Of course it's feasible. Everything (well, just about everything) that was feasible in the 1970s and 1980s is not only feasible today but actually MUCH easier. 

You can make LC oscillators. The capacitors and pot cores are easily available. Capacitors with a negative temperature coefficient (to compensate for the potcore tempcos) might not be so easily available. VCOs that cover the audio range are easy to make with op-amps 

Generally the simplest method of synthesizing sounds consists of generating some sine waves, and some noise (a zener diode can generate noise) and filtering the noise and gating the sounds through some envelope generators that have specific attack, decay and sustain characteristics. Something like a cymbal would have a bunch of noise, a piano much less so. 

But just because it's feasible doesn't mean that just anyone can do it. They could always find an old construction article in dead-tree format and attempt to adapt it even if they were not capable of doing the design. Probably most of the stuff is still available. 

Unlike back then anyone can get boards made cheaply and reasonably quickly, so assembling some hundreds of parts is not such a big deal. 

This is not going to be something that sounds like a modern electronic instrument that uses DSP and sampling techniques, but something that sounds like something from a foregone era, and with annoyances (or perhaps charms) such as frequency drift and so on. 

Anyway, I'm sure if you troll through the archives of the Brit publication Practical Electronics and similar publications you should find some inspiration.

You could start with the Russian invention Theremin? Not much circuitry there.