Experimental result of the atomic nucleus volume by scattering alpha particles from the atomic nucleus. Investigation of the electron volume by what? Rutherford's alpha particle experiments marked the beginning of the determination of the volume of the atomic nucleus.
How were the experiments conducted that led to the statement of the point-like electron?
 A: 
How were the experiments conducted that led to the statement of the point-like electron?

The particle physics experiments are mainly scattering experiments, and the theory that fits the existing data and predicts future data is the quantum field theory, QFT, of the standard model, SM.
The axiomatic table in the SM has all the particles point like in the QFT, i.e. for each individual vertex in the Feynman diagrams the particles are considered as points in the integration, no extent or complexity. The fit of theory to data is very good, this means that it is not necessary to change the axioms.
Now the complete calculation is different than the first order diagram, each addition of higher order diagrams brings corrections and if one looks at the "shape" of the electron it will not be a point but a locus in probability space
But the experimental evidence for assuming the vertex particles pointlike comes from the successful fit of the SM to data, up to now. Anyway, experiments are running in order to find disagreements with the SM, who knows what the future will bring?
A: To enlarge slightly on Anna V's answer, here is what we mean by scattering experiments.
Let's imagine a tight beam of electrons zooming through a vacuum, aimed at a target consisting of a simple sort of atom, and separated into bunches of electrons. So... we are going to "machine-gun" the target atom with "bullets" consisting of electrons.
Then we surround the target with sensors that tell us at what angles the electron bullets bounce off the target atoms. This "ricochet" data tells us how big the target is and what shape it has, if any.
At low electron energies, the electron bullets bounce off the electrons that surround the target atoms, and we find out how big the atom is. As we increase the energy, the electron penetrates the electron cloud and bounces off the nucleus inside the cloud, and we find out how big the nucleus is. At higher energies, the electrons bounce off of individual particles (protons and neutrons) inside the nucleus, and now we know the diameter of a proton or neutron. More energy, and we start bouncing electrons off the quarks inside a single proton or neutron.
Although this is a simplified picture, in all cases the ricochet data tells us that the electron doesn't have anything "inside" it (unlike a proton, which has a well-defined diameter and internal structure in the form of three little quark guys running around in there) and that its own diameter is far smaller than anything we can measure with scattering experiments like this.
