What do historians do when there are no sources?

Historians interpolate meaning from multiple conflicting textual sources in the documentary record of the past. This is the natural behaviour of the historian. Between a newspaper article on Thursday and one of Friday the historian must simulate the occurrences of the intervening day, and then imagine that totality of "Thursday" and how it would impact on Friday's news. This is true even with the most comprehensive, diverse and complete documentary record. Historians produce an imaginary interior built out of multiple sources and perspectives. When an interpretation is no longer solid "A clown invasion made the Jury declare Guilty." this tends to become obvious, as the interpretation becomes tendentious, rests on fewer text points or more obscure interpretations, and also "just doesn't make sense" from the perspective of other historians simulated imaginary. Historians, therefore, tell rhetorical stories to try to make sense of what they imagine based on what they read. The important thing about history is: they try to make the story true to the past as it was, rather than their desires about what the past ought to have been.

This governs the rest of the answer.

Historians stitch together diverse but inconsistent series by knowing how different sources will talk about a common thread or process. Courts talk about things in one way with one set of limits, starvation chronicles talk about something in another way. If we have a common point where we can see how starvation chronicles talk about something when people are being tried for hoarding, we can stitch the rest of the story together knowing the limitations on the diverse sources.

Where a source is consistent but limited, we go to other instances. For example, if the Courts in Scotland rarely encounter women in the middle ages, and the Courts in England rarely encounter women in the middle ages, but in England we have other sources on the history of women; then we use the difference between the Courts' partial reporting on womens' lives and the fuller story in England to hazard an interpretation of the probable limits of Scottish Court sources.

Where a series is partial and inconsistent, we try to produce metaphoric accounts from other societies ("theory"), and then apply the theory with intense and incredible scrutiny of individual sources. By more closely reading the limited series, we push the interpretive capacity to the limit to produce what can be gleaned from the sources. When there is less and worse material to read, we read harder. We also start using non-documentary records of the past, such as archaeological, anthropological, literary, religious records. We start to stop being historians, and become interdisciplinary scholars.

Where only a singular textual source exists we can only comment on that source. We probably should stop considering this as history, but, occasionally we can uncover a context (say through linguistics) and then relate this source to other sources. Singular sources are a problem of finding their appropriate context.

Where textual sources are unreadable (Linear A) we cannot be historians. We must become, perhaps, historical archaeologists, or archaeologists with a historical interest.

Where no documentary record of the past exists history is impossible. Other scholars (anthropologists, archaeologists) can provide information about the past. Similarly, it may be possible to interrogate the past through oral records of the past that are actually a documentary record if you think about it without Western bias. In circumstances without a documentary record, sometimes further work on methodology and theory can uncover that "actually yes, there was a documentary record all along but we were too blind to see it." Also, often, in this case historians may be waiting on information professionals to supply a documentary record. Until the "cabinet" papers open, most political history lacks the substantial basis for its understanding. Post-soviet history of the Soviet Union is a wonderful field, as the archives opened.