Modern science has replaced the standard platinum-iridium reference meter (m) stick with the indirect prescription, ". . . the distance travelled by light in empty space during a time of 1/299,792,458 of a second," where a second (s) is now defined as the time it takes a certain frequency of the light emitted by cesium atoms to oscillate 9,192,631,770 times.3.2 This represents a significant improvement inasmuch as we no longer have to resort to carrying our meter stick to the International Bureau of Weights and Measures in Sèvres, France (or to the U.S. National Bureau of Standards in Boulder, Colorado) to make sure it is the same length as the Standard Meter. We can just build an apparatus to count oscillations of cesium light and mark off how far light goes in 30.663318988 or so oscillations [well, it's easy if you have the right tools . . . . ] and make our own meter stick independently, confident that it will come out the same as the ones in France and Colorado, because our atoms are guaranteed to be just like theirs. We can even send signals to neighbors on Tau Ceti IV to tell them what size to make screwdrivers or crescent wrenches for export to Earth, since there is overwhelming evidence that their atoms also behave exactly like ours. This is quite remarkable, and unprecedented before the discovery of quantum physics; but unfortunately it does not make much difference to the dilemma we face when we try to define "distance." Nature has kindly provided us with an unlimited supply of accurate meter sticks, but it is still just a name we give to something.
To learn the properties of that "something" which we call "distance" requires first that we believe that there is truly a physical entity, with intrinsic properties independent of our perceptions, to which we have given this name. This is extremely difficult to prove. Maybe not impossible, but I'll leave that to the philosophers. For the physicist it is really a matter of æsthetics to enter into conversations with Nature as if there were really a partner in such conversations. In other words, I cannot tell you what "distance" is, but if you will allow me to assume that the word refers to something "real," I can tell you a great deal about its properties, until at some point you feel the partial satisfaction of intimate familiarity where perfect comprehension is denied.
How do we begin to talk about time and space? The concepts are so fundamental to our language that all the words we might use to describe them have them built in! So for the moment we will have to give up and say, "Everyone knows pretty much what we mean by time and distance." This is always where we have to begin. Physics is just like poetry in this respect: you start by accepting a "basis set" of images, without discussion; then you work those images together to build new images, and after a period of refinement you find one day, miraculously, that the new images you have created can be applied to the ideas you began with, giving a new insight into their meaning. This "bootstrap" principle is what makes thinking profitable.
Later on, then, when we have learned to manipulate time and space more critically, we will acquire the means to break down the concepts and take a closer look.