The previous answer is not correct, so I thought I'd correct it. While we often phrase naturalness in terms of the size of coupling constants, as these are what we usually work with mathematical, we could rephrase everything in terms of physical measurable quantities and the same considerations would still arise. So the issues surrounding naturalness have nothing to do with our choice of how to describe the physical system, contrary to the previous answer.
The underlying rationale behind Dirac naturalness is the observation that if you have a physical system with some length scale $L$, then we should expect all physical quantities, once suitably made dimensionless, to be $O(1)$ with respect to that length scale. What $O(1)$ actually means isn't that clear, but I think most model builders would consider the gauge couplings and the Yukawas of the third generation of matter (which are $\sim 10^{-2}$ or larger) to be Dirac natural in this sense. Whether we should consider the Yukawas for the second and first generation of matter (which can be as low as $\sim 10^{-5}$) to be Dirac natural is less clear. The Higgs mass ($\sim 10^{-17}$) and the cosmological constant ($\sim 10^{-30}$) are definitely not Dirac natural, and so considered problematic by most (but not all) model builders.
Technical naturalness is a broader concept than Dirac naturalness. A quantity is technically natural if it does not suffer from large quantum corrections. The reason we might care about technical naturalness is that it allows us to punt our explanation for why a quantity is small to higher energy scales. A simple example is the neutrino mass. This is technically natural, because the neutrino has a chiral symmetry when it is massless. But one cannot tell if the neutrino mass is Dirac natural just by studying physical process with energies comparable to the neutrino mass. Instead, we have to appeal to energy scales much greater than the neutrino mass itself. In the Standard Model this is realised through the seesaw mechanism, which is Dirac natural. (Actually, in the Standard Model this isn't Dirac natural because it relies on the Higgs mass being light. However, it doesn't introduce any new hierachies beyond the usual Higgs mass problem. Hypothetically, you could realize light fermions through a strongly coupled gauge theory breaking a chiral symmetry, and in this case you would have a Dirac natural explanation.)
Of course, it is possible that there are technically natural explanations that can never be realized through a Dirac natural explanation at much higher energy scales. If this is the case then we should rule out these technically natural explanations. But in practice it is hard to know for sure whether this is the case (maybe we have simply not found the Dirac natural explanation yet), so I think this is why model builders focus on technical rather than Dirac naturalness.
This also explains why model builders are much more interested in the Higgs mass than the Yukawas in the Standard Model. Aside from the much greater fine-tuning required by the former compared to the latter, the Yukawas in the Standard Model are technically natural, and so maybe GUT or Planck scale physics provides a Dirac natural explanation for their small size. The Higgs mass is not technically natural, and so if we want our physics to be Dirac natural we must invoked new physics at the weak scale.
To summarize: The absence of technical naturalness implies that the lack of naturalness needs to be solved at roughly the energy scales the problem arises if we want the theory to be Dirac natural. But if a quantity is technically natural, we may not be able to tell if it is Dirac natural without understanding physics at much higher energy scales.
