How can one make an analog voltage memory circuit?
This is not a practical answer unless you happen to work for a company with the resources of, say, Intersil, but the technology exists to make this work. Consider the ISL21080 type references which hold charge, hopefully for the life of the equipment they're installed into, based on a tiny capacitance isolated by quantum tunneling effects. Provided they don't get too much in the way of X-rays etc. they'll remain pretty stable for years. See, for example, this application note. 



I might add this kind of thing gives me the willies. 

For an ordinary application, digital is most likely the way to go.