How can MRI overcome diffraction limit? MRI uses radio waves with lengths of meters. 
For example, Larmor frequency for proton in the field of 1 Tesla is $42~\mathrm{MHz}$. This means $7~\mathrm{m}$ wavelength.
So how can MRI scanner see details of 1 millimeter of size with the waves of $7~\mathrm{m}$ length?
 A: As with all imaging that "overcomes diffraction limits" (e.g. also for optical superresolution) the imaging technique makes use of knowledge aside from the sensed radiation to locate the source of that radiation.
In medical MRI, the human body is subjected to a magnetic field that varies with position such that only a small region within the body can absorb the RF excitation energy. That is, only the protons in a small region in the body have a Larmor frequency that matches that of the excitation RF. Then the excitation frequency is changed and / or the gradient of the magnetic field is changed so that a new region of the body resonates. We know from the applied magnetic field's magnitude where in the body the MR radiation comes from. We only need to detect the magnitude with the RF receiver apparatus to make the MRI map - we don't locate the source with RF receiver.
Other methods that use this principle to cement the idea:


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*Two-photon imaging. The two-photon absorption cross section is proportional to the square of the excitation light field intensity. Therefore, if a scanning, focussed field in a sample leads to two-photon fluorescence, then we only have to receive the fluorescence to make an image. We don't have to locate the source with the receiver optics, because, owing to the square intensity dependence, it is overwhelmingly probable that the fluorescence came from the focal region of the excitation beam.

*Optical superresolution There are many schemes, but one example is Stefan Hell's Stimulated Emission Depletion microscopy idea. Before gathering light from a focal region, the excitation light is extinguished. But before fluorescence is gathered, fluorophores near the focus are selectively deactivated through stimulated emission by a pulse of light with a doughnut shaped focal region. This means that when fluorescence is gathered, we know that it can only have come from a certain region - that within the "hole" of the doughhnut, and we know where that region is to a tolerance that is less than the diffraction limit.
