It is best to answer these questions in reverse order.
So, do elementary particles have a size at all? Well, elementary particles are called elementary as they are the most basic constituents. In our theories today, they start out as pointlike. Only particles made from other particles, so-called bound states like a nucleus or a hadron, have a size. And now comes the but.
First of all, we do not yet know whether our elementary particles are really elementary. They may also be bound states of even more elementary particles. But in experiments we can only determine upper bounds to the size. Making better experiments will reduce this upper bound. Eventually, we may see that a particle previously thought of as point-like has a size. This has happened quite frequently over time. It always opened up a new level of elementary particle theories. Therefore measuring the size is important. But for us, as theoreticians, this type of question is only important if we have an idea about what could be the more elementary particles. And while some of our research is going into this direction, this project is not.
The other issue is that quantum effects give all elementary particles an 'apparent' size. This comes about by how we measure the size of a particle. We do this by shooting some other particle at it, and measure how strongly it becomes deflected. A truly pointlike particle has a very characteristic reflection profile. But quantum effects allow for additional particles to be created and destroyed in the vicinity of any particle. Especially, they allow for the existence of another particle of the same type, at least briefly. We cannot distinguish whether we hit the original particle or one of these. Since they are not at the same place as the original particle, their average distance looks like a size. This gives even a pointlike particle an apparent size, which we can measure. In this sense even an elementary particle has a size.
So, how can we then distinguish this size from an actual size of a bound state? We can do this by calculations. We determine the apparent size due to the quantum fluctuations and compare it to the measurement. Deviations indicate an actual size. This is because for a real bound state we can scatter somewhere in its structure, and not only in its core. This difference looks pictorially like this:
So, do we know the size already? Well, as said, we can only determine upper limits. Searching for them is difficult, and often goes via detours. One of such detours are so-called anomalous couplings. Measuring how they depend on energy provides indirect information on the size. There is an active program at CERN underway to do this experimentally. The results are so far say that the size of the W is below 0.0000000000000001 meter. This seems tiny, but in the world of particle physics this is not that strong a limit.
And now the interesting question: Why do we do this? As written, we do not want to make the W a bound state of something new. But one of our main research topics is driven by an interesting theoretical structure. If the standard model is taken seriously, the particle which we observe in an experiment and call the W is actually not the W of the underlying theory. Rather, it is a bound state, which is very, very similar to the elementary particle, but actually build from the elementary particles. The difference has been so small that identifying one with the other was a very good approximation up to today. But with better and better experiments may change. Thus, we need to test this.
Because then the thing we measure is a bound state it should have a, probably tiny, size. This would be a hallmark of this theoretical structure. And that we understood it. If the size is such that it could be actually measured at CERN, then this would be an important test of our theoretical understanding of the standard model.
However, this is not a simple quantity to calculate. Bound states are intrinsically complicated. Thus, we use simulations for this purpose. In fact, we actually go over the same detour as the experiments, and will determine an anomalous coupling. From this we then infer the size indirectly. In addition, the need to perform efficient simulations forces us to simplify the problem substantially. Hence, we will not get the perfect number. But we may get the order of magnitude, or be perhaps within a factor of two, or so. And this is all we need to currently say whether a measurement is possible, or whether this will have to wait for the next generation of experiments. And thus whether we will know whether we understood the theory within a few years or within a few decades.
