This question has bugged me for a while as it's not addressed in any books I've read. I was wondering how folks from this community interpreted it. Below are my personal notes on the matter (please excuse the formal tone).
In the context of signal sampling, one cycle of an event represents a single, complete, sampling (measurement) of a value of a quantity. The interpretation of (positive) integer sampling rates is intuitive: fs = 5 Hz means 5 samples are taken every second. But how to do we interpret fractional sampling rates (e.g., fs = 1/3 = .333... Hz)?
By definition, a sampling frequency of 1/3 means one-third of a sample is taken every second -- but what does one-third of a sample mean? The idea of a third of a sample of a measured value is indeed absurd (is the measured value "a third-valid" and "two-thirds invalid"?) but one way to make sense of it is to consider the imperfect nature of a sampling operation.
Though we can conceive of sampling as an operation that occurs instantaneously in the abstract, we must realize that all system takes some non-zero amount of time to acquire a value of a quantity (a delay that manifests as a physical limitation of the measurement procedure or is introduced by hardware that carries out the analog sampling).
So it becomes conceivable that a unit of time (one second) might not be enough time for a single sample of a quantity to be available (or perhaps only one-third of the sample is available every second). We can interpret fractional frequency as an incomplete sample per unit time or the fraction of a sample that is available per unit time.
Another equivalent (perhaps more sensible) way to interpret a fractional sampling frequency is to interpret its mathematical inverse, the sampling period, which is the number of seconds required for a single, complete measurement of a value of a quantity.
Edit: Not sure why I was downvoted. I think this is a legitimate question. Please specify how I can improve the question.