How might I get hold of a dataset from an old paper? Say I come across a paper e.g. from 1980, which refers to a dataset which I believe could have further interesting information that was not analysed (or at least the analysis was never published). This was long before the time when datasets might be included as supplementary information. The sole author has not published recently (10 years +) and I can’t find any reference to them at the listed institution or indeed anywhere else. They are likely retired, quite possibly passed away by now.

Given that the author is no longer contactable, is there any way of getting hold of that old dataset? My instinct is that the answer is “no”, but is there any hope? How could I go about it?
Lots of old datasets still exist and are stored in repositories, exactly to solve the problem you face. So my first port of call would be to look there. Which one depends on your discipline.

If that is not the case you can try track down her or his PhD students from that time. Chances are they worked with that data, and may know what happened to it. 

Sometimes data is just lost. I remember a case where data was stored on punch cards in a university cellar, and there were also mice that liked eating cardboard in that cellar...