Photons, for example, certainly exist. Look at the glowing CFL above your head.
“They [photons] carry momentum but have no mass IF AT REST - IN FACT, THEY CANNOT EXIST AT REST(though their phase velociy can be slowed to be near at rest).”
This statement is from a scientific forum somewhere on the web (emphasis mine). So, I admit my non-physicist brain struggles with the concept of a “massless particle” as anything but a theoretical construct, just as I struggle with the idea of the universe expanding into something - some hypothetical space - that has (as you put it) “no operable definition.”