Tom Stoer hit the nail on the head. When asking, "How large are electrons?" one is really asking, "How do electrons behave in situations where particles of different sizes behave differently?" If you can't come up with an experiment to demonstrate the concept, it's quite possible that the concept itself is physically meaningless.
How large is a bowling ball? About 20cm across. How do I know? Because I have some square blocks that are 4cm tall, and when I stack five of them they reach the same height as the bowling ball. Great! Can I do that with electrons? Unfortunately, no. I expect neutrinos to be smaller than electrons (because they end with "-ino," which sounds smallish) but I can't come up with a way to line them up end-to-end. That sucks. Oh, but here's a better idea! If I stick five bowling balls next to each other, they form a line that's one meter long. Can I stick a bunch of electrons in a row and measure the length of the row with an ordinary ruler? Hrm, no, lining up electrons end-to-end doesn't seem to work, either.
OK, I'm desperate, so let's try an approximation. If I have a gigantic hollow cube, 1 kilometer on each side, and I fill it with bowling balls and then count the balls. There are 216 billion. How big is a bowling ball? Well, if bowling balls are spheres and you pack them with minimal wasted space, they'll fill 90.7% of the available space. (The rest is gaps between balls.) So of the billion cubic meters of space, we've filled 907 million cubic meters with bowling ball; divide by 216 billion balls to get 4200 cubic centimeters each. Now V = 4/3 pi r^3 for a sphere so the ball's radius must be 10cm, so its diameter must be 20cm. Can I do anything like that with electrons? Counting electrons is easy because they're charged particles. So how many electrons can I pack into empty space? And is that number limited by the size of the electron? (If it isn't, then this argument is useless.)
The vanilla model of quantum mechanics says you can pack as many electrons as you want into any size of space, provided you have enough energy. (The electrons repel each other, and there are a lot of electrons in this block of space.) So in that sense, the electron could be imagined as having no volume. But even then, this way of looking at things is total bunk, because nothing prevents two electrons from having nonzero amplitudes to be found at the same spot, just so long as the amplitude that they'll both be found at the spot simultaneously is zero. It turns out that electrons, like all particles, aren't really localized into one position in space. There's a rule that says no two electrons can have the same "state" at the same time, but when you break it down you see that the rule doesn't really say anything about physical overlap. So this approach to measuring the electron radius just doesn't work.
OK, let's try something clever. I happen to know that my bowling ball is 7kg and when I grab it and try to make it spin, it takes 1.4 Joules to make it rotate at 10 radians per second. Assuming that it has uniform density (which isn't true in general; most bowling balls are denser towards the middle), we'd have its rotational kinetic energy given by KE = 1/2 * (2/5 mr^2) omega^2 with KE = 1.4 J, m = 7kg, and omega = 10rad/s^2, so we could get r = sqrt(5*KE/m)/omega = 10 centimeters. Awesome! Electrons spin, too, so we can just measure the kinetic energy of their spinning, the electron mass (which we know), and the angular rate at which they're spinning and we can get a mass, based on the assumption that they're uniform spheres.
Except... There's no way to measure a kinetic energy in an electron's spin, or the rate at which it's spinning. We can tell whether it's spinning "up" or "down" relative to some direction we've chosen, but there's no way to turn that into a physical rotation through a certain number of radians. And the only energy we've been able to measure that's related to spin is actually a measurement of the interaction of spin with an ambient magnetic field. So the moment of inertia argument doesn't really work, either. And even if it had, how would we have been able to show that the electron has uniform density?
Eventually, someone came up with an ingenious argument that *could* yield an answer. Relativity says that particles have a rest energy: E = mc^2, which for an electron comes out to 8.187*10^-14 Joules. What *is* that energy? Well, imagine a bunch of negatively charged dust scattered "infinitely thinly" across the universe---it has energy zero. Try to collect that dust into a single particle. It's negatively charged, so all the little dust motes are repelling each other as you bring them together. Thus it takes energy to pull them together. Eventually, after considerable effort, you have assembled them into a little spherical shell---an electron. It probably took you 8.187*10^-14 Joules of energy to assemble this particle, since that's the amount of energy in an electron... and an electrostatic argument says that it should have taken you E = 1/2 * (1/4*pi*VacuumPermittivityConstant) * (total charge)^2 / (radius of shell). The total charge is the charge of the electron, the vacuum permittivity constant can be found in a book, and the charge of an electron is easy to measure... That leaves the radius of this shell, which must be the radius of the electron. Great! Do the math---it comes out to about 2.818 femtometers.
So now we have the radius of the electron! Except we don't. The electric field doesn't work this way in quantum mechanics, and since we know that the world is *really* quantum mechanical, we'll have to throw this out and go back to the drawing board. Darn it.
Eventually we have to give up. Quantum mechanics doesn't give us any good experiments that will give us a length that can rightfully be called "the radius of the electron." The closest we came was with a classical argument about building an electron out of negatively charged dust, and even that was indirect. (Nobody actually builds electrons that way.) Someone may come along with string theory and say, "Electrons are really tiny loops of string and the loops have such-and-such radius." But don't let that fool you---they don't mean that if your finger were smaller than that, you could fit it inside the loop. They actually have another indirect argument---a vibrating string resonates at certain frequencies. The allowable frequencies depend on the length of the string. If you can find a relationship between the frequency of a vibration and the amount of energy stored in it, and then you can measure the allowable energies, you can figure out what length the string had. But that's hokey. Saying that the particle is really a piece of string is like saying you can build an electron from a fine negatively charged dust... It's a nice thought experiment but you'll never really be able to do it directly. So the number you get when you're done, though it may provide some useful insight for certain problems, might not actually correspond to anything "real."
Why didn't I just say that from the beginning? Because by exploring all of the failed attempts to determine an electron radius, we got to see a lot more about how physicists have to operate.
