Determining mass radius and charge radius of electrons First the mass radius problem:
Why can't the mass radius of electrons be determinded by shooting neutral particles on it. Similar to Rutherford's gold model only a bit  more sophisticated. 
Secondly the charge radius problem:
I often hear the term charge radius and the charge radius for protons has been calculated with accuracy for example described in this article:
https://phys.org/news/2016-08-deuterium-nucleus-proton-radius-puzzle.html
Why can't the charge radius of electrons be determinded in a similar way?
 A: Rutherford used $\alpha$-particles as projectiles. Compared to the gold foil they have "little" mass and they interact "heavily" via the electro-magnetic force. Furthermore, compare to the size of a gold atom (incl. the electronic shells) the $\alpha$-particles are small. All three properties were important:


*

*small mass, so that the gold foil will not move when it interacts with the $\alpha$-particles. Otherwise, we would not find back-scattered particles.

*"strong" interactions via the electro-magnetic force, so that we can find some reflected particles in the forward direction.

*small size, so that it forms a local probe.


So what neutral particles would you use in the electronic Rutherford experiment?
A: Mass radius refers to overall neutral charged particles like regular not ionized atoms. Nuclei (with the exception of He-4 nucleus) and electrons are charged particles and their radius is their charge radius.
Moreover, electron particles are considered by the SM as dimensionless point mass-charge particles meaning they are considered effectively as not having any radius. This is because the Standard Model (SM) as an effective theory has adopted the "bare" mass or else "naked" model of the electron where the electron is described like a dimensionless center of mass type of point in space from where intrinsically all its properties originate like mass, charge, spin etc., opposite to the more physical charge "dressed" model of the electron which describes the electron with its charge. So really all the experiments and research involved today to define a finite radius for the electron, fall out of the scope of the SM and beyond and must be characterized as New Physics.
Of course SM as an effective theory is not carrying about what is actually there and only what is necessary for describing accurately the results and outcomes of phenomena and make predictions. It is more a quantitative and less a qualitative theory.
With that said as far as I know you cannot perform any light scattering measurement on free electrons because the elastic scattering of the incident photons on the electrons thus the kinetic energy of the photons is preserved after scattering and there is no wavelength shift that would permit any meaningful data to be collected for making calculations.
You can only perform inelastic scattering (Compton or Raman scattering with X-rays of gamma radiation) on bound or loosely bound state electrons inside the atom but then again you end up measuring electron orbital shells and not really the charge radius of a free electron at rest which is the most interesting we want to learn if the electron has a finite radius.
There are some theoretical calculations of the upper limit of the free electron at rest derived from measurement of other characteristics of the electron like its g-factor using a Penning trap or pure theoretical like this here for the upper radius limit but they don't agree and there is currently no direct method for measuring the radius of a free electron. This is a secret nature holds very dearly.
The best indication we have assuming quarks have the same size as free electrons, is this HERA team experiment of bound quarks inside the protons and neutrons with the upper limit of the charge radius set at $0.43 \times 10^{-18} m$ and also here from ZEUS team with upper limit $0.85 \times 10^{-18} m$ but again it is also a matter of what you define as radius including also the dressing virtual charges and so on.
Further trying to measure the charge dimensions of the electron with electrostatic or magnetostatic methods is in vain since these are the interaction far fields of the electron with the environment generated by the electron charge, and do not represent the near field of the electron thus its charge which is a self-contained and space confined energy field.
Bottom-line, we don't have a clue what the finite radius of a free electron at rest is and there is no method currently to directly measure this using light scattering or any other method.
Note: I find the following quoted text and reference as an interesting reading about this matter:

From the orbital model of the structure of elementary particles, the electron is structured by the orbital of an integer negative electric charge, orbital whose size is of 2.82 Fm (i.e Classical radius of electron). However, it is often believed that the electron is punctual. This is due to the fact that in high energy collision experiments its structure is not detected, being very light (only 0.51 MeV) in front of the collision energy and so its point-like appearance come from its structural electric charge which is punctual (less than 0.001 Fm) [1].

A: One reason why a Rutherford-like experiment would't work in the case of the electron is because electron is an elementary particle i.e. it is not made of something else, so shooting particles at it won't have any effect in this sense (of course you can create another particle by colliding for example an electron and a positron, but this will not give you information about the structure of the electron). On the other hand, a proton is made of quarks, so you can do a Rutherford experiment in order to see it's inner structure (and it was done with a very high accuracy). However, in both cases, it is hard to define a precise radius, as both particles have a wave-like behavior so you can define a region in space where you can find the particle with 99% (let's say) certainty, but you can't define it as a solid sphere with a certain radius.
