What is antisymmetric exchange? What is J-strain? Where does it come from?
Maybe OP has already contacted the authors of this paper by now, but this was interesting. I'm no expert but I can Google things and this was too long for a comment.

Antisymmetric exchange:
At first I thought it was simply an exchange interaction where the wave function's sign is changed during exchange, now I don't think it's so simple.

Antisymmetric exchange is also known as DM-interaction (for Dzyaloshinskii-Moriya). I don't know exactly what it is, here is the original paper citation - can't find it anywhere though. 

I.E. Dzyaloshinskii, Thermodynamic theory of weak ferromagnetism in antiferromagnetic substances, Sov. Phys. JETP 5(6), 1259-1272 (1957)

A definition

A slideshow, which contains some more info (slides 16-23)

J-Strain: Reading this, "J-strain" isn't the only "strain" that's referenced in this paper. Also referenced within is "G-Strain" which has a more elucidating Wikipedia article.

So, maybe it's safe to say, like G-Strain, J-strain is the broadening of the "distributed exchange coupling constants" (a.k.a. J-values). Inferred from a line in the paper: "Both proteins exhibit distributed J-values (J-strain)".

What did J-strain have to do with the experiment? "...The observed J-strain, which presumably represents a distribution of protein conformations “locked in” upon freezing the samples."

Where does the broadening of the J-values (exchange coupling constants) a.k.a. J-strain come from? Well apparently there's this thing called J-coupling that seems relevant. J-coupling is also known as indirect dipole-dipole coupling, and gives info about molecule interconnectivity. J-coupling relates nuclei to local electrons (All this is from the J-coupling article). The broadening of the lines may come from anisotropic interactions, which affect nuclear spin energy levels. Sure enough, the paper says it can't rule out anisotropy. An Alternative source? Non-homogeneous regions in the sample, which could go back to the frozen/locked conformations.

I still think it'd be great if someone doing physics research in this field could elucidate some more though.

Another J-coupling Webpage (U. Wisc)

Of course I should cite the original paper, "Evidence for Antisymmetric Exchange in Cuboidal [3Fe−4S]+ Clusters". Linked in question.