What really enforces technical naturalness of electron mass? Technical or 't Hooft naturalness A parameter $\theta$ in the Lagrangian of a field theory is said to be natural, if in the limit of vanishing $\theta$, the theory has some enhanced symmetry. If this happens, the smallness of the parameter $\theta$ is said to be natural.
An example Let us consider the theory of QED with massless electrons. It can be shown that the electron mass does not receive quantum corrections and remains zero. If we repeat the same calculation with massive QED, we find that the bare electron mass $m_0$ receives a correction which itself is proportional to $m_0$ i.e. $$m_0\to m=m_0+\frac{3\alpha}{4\pi}m_0\ln\Big(\frac{\Lambda^2}{m_0^2}\Big)\tag{1}$$ where $\Lambda$ is the cut-off. Thus, massive QED reproduces the result of massless QED in the limit $m_0\to 0$. This clearly shows that if $m_0$ is zero in the classical action to start with it will remain zero; if $m_0$ is nonzero but small to start with, it will remain small. In this sense, the smallness of electron mass is technically natural.
Question But I am still uncomfortable with the role of symmetry here and cannot fully digest the idea of technical naturalness. Because, if the symmetry is anomalous in the limit $m_0\to 0$, what is it that stabilizes the electron mass against large quantum corrections? Since the symmetry is anomalous, we cannot say for sure that it is the symmetry that protects $m_0$ from receiving large correction. What is really going on at the heart of the matter?
 A: To forbid this mass term, we do not need a full chiral U(1) symmetry, but only a $\mathbb{Z}_2$ symmetry under which, say, the left-handed electron picks up a sign. This $\mathbb{Z}_2$ is anomaly free and can thus forbid the mass term also at the quantum level.
A: In a general chiral gauge theory in D-dimensions, the anomaly only appears in the one-loop diagram with (D-1) external gauge bosons (arXiv:0802.0634). 
The chiral anomaly you are talking about only appears in the triangle diagram (in 4 dimensions), i.e. 3 external photons with an electron loop. Adler, Bell and Jackiw proved this perturbatively, but a more modern treatment can be found in arXiv:0802.0634.
So the loop corrections to the electron mass don't break the chiral symmetry. Only the triangle diagram does.
The non-conservation of left- and right-handed fermions only happens with non-zero electric/magnetic fields, since the divergence of the chiral current is proportional to the field-strength tensor.
See chapter 19.2 of Peskin & Schroeder, as well as problem 19.1.
I'd also recommend following Fukikawa's analysis, where the anomaly comes from the Jacobian of the field measure in the path-integral. Also done in P&S.

EDIT: I've neglected non-perturbative effects in this answer (which is okay for the photon, since abelian gauge groups have no topological configurations). See the comments section.
If you want to learn more about calculating such effects (saddle-point expansion/semiclassical method) I'd recommend https://iopscience.iop.org/article/10.1070/PU1982v025n04ABEH004533/meta
A: It's not just the chiral anomaly that can muddle the argument that chiral symmetry protects masslessness.  If we start with  massless quarks the strong interactions would still cause chiral symmetry breaking and give the quarks a "constituent" mass.  Indeed in the real world the "current" masses of the the $u$ and $d$ quarks are rather small and most of the masses of the  hadrons containing them comes from chiral symmetry breaking and so are  proportional to $\Lambda_{\rm QCD}$ rather than $m_u$ or $m_d$. 
