There are many problems in the pursuit of an accurate measurement of the size of hte Universe, both theoretical and practical. I will discuss these (as I understand them).
THEORETICAL
-The first problem must be deciding what, specifically, we mean when we say the size of the Universe. Imagine the Universe is spherical in shape (unlikely, but easy to visualise): do we mean to measure the diameter of the Universe, or perhaps the diameter? Either way it is unlikely that the Universe is as simple as our own planet, as theorists do not believe the Universe is a perfect sphere, but perhaps a complicated geometric shape. (There is also the strong possibility of the Universe containing many more dimensions that we cannot detect, and whether we should be taking the size of these dimensions into account when measuring the size of the Universe.)
-The second problem is identifying the boundaries, the edges, of the Universe. The fact is we cannot observe them due to the vastness of the Universe is one issue; another is that scientists believe that the Universe may be finite but never ending! (Remember PacMan? There were some versions of the game where you could exit a specific point on the right hand side of the screen and magically re-enter on the left hand side, and vice-versa; same on the top and bottom. Some speculate that our Universe is like that, a finite space but one we could travel forever without reaching the end.)
PHYSICAL
-Having decided where the Universe ends and what we want to measure we then have the easy task of measuring this! Sadly we do noy have a big enough ruler, so that won’t work! To measure distance to the Sun and Moon we can send a radio signal, which rebounds from the physical surface of the body, and we can then detect the return signal. Using the time between sending and receiving this signal, and knowing the speed of light (which is a universal constant) we can easily calculate the distance the signal has travelled, and so how far away the body is. This obviously doesn’t work because there is nothing to bounce the signal off at the end of the Universe.
-The fact that the Universe is still expanding (that is to say the distance between galaxies is increasing) also means that any physical measurement of this will be relevant and correct only for a particular time.
CONCLUSION
It is likely that is impossible to physically measure the size of the Universe, and that the only ‘measurement’ of this will come from theoretical models, sculpted by massive supercomputers. Unfortunately, even at the present time, we know far too little about the Universe to be able to accurately model it. Currently we cannot even measure the age of the Universe, nor its rate of inflation (expansion), and we would certainly need both of these (and much more) to predict the size of the Universe.
