"A line segment can be divided, at least in thought, ad infinitum by halving it, halving the halves, and so on without end. So it must be made up of infinitely many parts. What is the size of these parts? If it is zero the line would have no length, if it some non-zero size, however small, the segment would have to be infinitely long."From Paradoxes from A to Z by Michael Clark (Routledge, 2002)
Although the apparent absurdity here may appear to be the consequence of this being a purely conceptual problem (i.e. such a state of affairs could not exist in the real world, and is merely a philosopher's fancy) there is a valid sense in which the conundrum applies in real world situations too.
Think of a real line, drawn on paper or screen (like the one above). It appears to be of fixed and determinate length — something that would fit comfortably inside, say, and A4 sheet of paper. It appears such a line could be readily measured, but this is true only up to a point. If one zooms in on either end in order to locate precisely where it finishes, there is a problem insofar as the end will be 'fuzzy'. Whether made of ink or pixels, whatever material the line is constructed from will ultimately be composed of atomic and sub-atomic particles. Such particles, at the quantum level at least, have no absolutely fixed location. Indeed the more one zooms in to find the precise point where the material constituting the line disappears the less determinate the point will be. In other words, there is a valid sense in which the precise length of a physical line is indeterminate, literally of in-finite length, although it may fit easily inside a sheet of A4 paper.
