What about acoustic delay line memory?
It is at least as "classic" as a drum but would be much easier to build.
AFAIK the CSIRAC was a bit-serial, delay memory machine.
I wonder if you could use those glass delay lines that were used in PAL
colour TV receiver until fairly recently (and there must be plenty still
around)? Alas they have a somewhat odd delay time (a little less than
64us -- one line period of the European TV signal), but I am sure you
could fiddle the master clock to compensate. There were some similar
dalay lines of exaclty 64us delay, used to store a picture line in the
drop-out compensator circuit of early VCRs, but those are going to be a
lot harder to find.
-tony