Where It All Goes Down
A non-Modern penal colony somewhere, in Modern Times
The larger setting of the story is very ill defined. We're in a penal colony somewhere "in the tropics," and that's about all we know. We don't know much of what the colony itself looks like – since practically all of the action takes place outside of the colony proper. We also don't know what sort of criminals are there, or where it's located geographically – beyond the fact that it's in the tropics and on the water (probably an island). No names are given to the colony, its location, or anything in it, including the characters (if we could even see what kinds of names they had, we might be able to get some idea – for example, "Olga" would indicate Russia or Eastern Europe, "Helmut" might imply Germany, etc.).
The native language of the colony isn't identified, though it is contrasted with the non-native French the officer speaks with the explorer. That the officer knows French quite well might make one wonder if French is the language of the elite of the colony, which could suggest Russia or Eastern Europe, where this was the case.
As for the time period of the story, it's probably sometime in the 19th century, given that the explorer is traveling on a "steamer." The apparatus seems perhaps more contemporary, seeing as it's a big machine running on batteries, but then again it's also a highly unlikely product of sheer fantasy.
What is perhaps most important about the colony, however, is that it is clearly not in Western Europe, nor could it be a colony established elsewhere by Western Europeans. Instead it's defined by opposition to "the West" (here too, the contrast between French and the native language is relevant). This opposition isn't just geographic, but also temporal and cultural; the West is seen as "modern" in a way the "traditional" colony, with its peculiar (or "outdated," for the new Commandant) tradition of justice established by the old Commandant is not.
We know this from the contrast the officer draws between the "Western" and "European" explorer who's visiting the colony and the colony itself. Both the officer and the new Commandant recognize that Western ideas of justice and humaneness are quite different from those of the penal colony. In this respect, the colony resembles any number of so-called "primitive" societies encountered by Western anthropologists or explorers (or imperialists) during the 19th century. We also know that this explorer is "touring the world" to visit many different societies and study their judicial systems. On the other hand, we also know from the officer that the colony is not "home" for the soldiers working there, and from their rather stuffy uniforms we might guess they come from somewhere not in the tropics. Beyond that, we can't really say any more.
As for the more specific setting of the story (the "microsetting"), all of the action, excepting the epilogue, happens at a particular spot outside of the colony itself, just as if it were a single-scene in a play. The only description we get of this spot is that it's a "small sandy valley, a deep hollow surrounded on all sides by naked crags" (1), where our four characters are gathered around "the apparatus." We also know it's quite hot, and the sun is very bright (which is a source of constant distraction for the explorer). In other words, it's a big machine in the middle of nowhere, a barren landscape.
Kind of an odd and unlikely image, isn't it? We'd say there's something pretty surreal about it. It's the kind of thing you might see in a Samuel Beckett play, which has the effect of making the story feel unreal or dream-like and pointing you more directly to questions about what it could mean.
Even weirder, since we know that the executions used to be very well attended, big ceremonial affairs: shouldn't there be a structure of some kind? The only indication we get that there's anything besides the machine is when Kafka mentions the mound of chairs that's apparently lying around near the machine.
With such an ill-defined penal colony and a surreal microsetting, it's not that hard to see the penal colony as some kind of symbolic or allegorical place – a place that represents something else. Check out "Symbols, Imagery, Allegory" for a possible allegorical reading of "In the Penal Colony."