Yes, it does make sense to ask, but the answer depends on what you mean by “size.”
A photon is the quantum of the electromagnetic field. In the current understanding of Quantum Field Theory, a photon is treated as an elementary particle, meaning it has no known internal structure and no measured physical diameter.
However, there are several different notions of “size”:
Classical size: If you ask, “What is the radius of the photon as if it were a tiny ball?”, experiments have found no evidence for any finite size. To the limits of measurement, it behaves as a point particle.
Wavelength: A visible-light photon may have a wavelength of about 400-700 nm. People sometimes loosely call this its “size”, but this is not really correct. The wavelength describes the spatial variation of the electromagnetic wave, not the dimensions of a little object.
Wave packet extent: A real photon can be prepared in a state spread over a finite region of space. For example, a photon emitted by an atom might occupy a wave packet several metres long, or even kilometres long if its frequency is extremely well defined. In this sense, a single photon can have an enormous spatial extent.
Interaction region: When a photon is absorbed by an atom or detected by an instrument, the interaction occurs at a localized place. This sometimes gives the impression that the photon was tiny all along.
So the question “What is the size of a photon?” is perfectly sensible, but physicists would usually respond by asking, “Which definition of size do you mean?”
Under our best current theories, the simplest answer is:
“A photon has no known intrinsic size or radius, but the quantum state describing it can extend over a region of space ranging from microscopic to macroscopic distances.”
This is one of the reasons photons seem so strange: they exhibit both particle-like localization and wave-like spatial extent, depending on how they are prepared and measured.