"self-resonance" sounds like not quite the way to describe the issue.
The problem is more simply that if the circumference is not very small relative to the wavelength, the antenna is no longer a small loop. As a rule of thumb, electrical engineers consider anything smaller than $\lambda/10$ to be "small". When things are "small" the analysis can be simplified with an approximation: current is equal everywhere in the loop.
There is no hard boundary where this approximation suddenly becomes invalid: in only becomes progressively worse. By $\lambda / 4$ it's certainly no longer a very good approximation.
As the loop becomes less "small", it becomes more like a folded dipole, and thus will have a substantially different feedpoint impedance and radiation pattern than a "small loop". It's possible to match such an antenna, but calculation of the necessary capacitance is different since the approximation of constant current is no longer valid.