We used K-band spectra to measure the H2 excitation temperatures in six molecular knots associated with the filaments in the Crab Nebula. The temperatures are quite high – in the range T∼ 2000–3000 K, just below the H2 dissociation temperature. This is the temperature range over which the H2 1–0 S(1) line at λ2.121 μm has its maximum emissivity per unit mass, so there may be many additional H2 cores with lower temperatures that are too faint to detect. We also measured the electron density in adjacent ionized gas, which on the assumption of gas pressure balance indicates densities in the molecular region nmol∼ 20 000 H baryons cm−3, although this really is just a lower limit since the H2 gas may be confined by other means. The excited region may be just a thin skin on a much more extensive blob of molecular gas that does not have the correct temperature and density to be as easily detectable. At the opposite extreme, the observed knots could consist of a fine mist of molecular gas in which we are detecting essentially all of the H2. Future CO observations could distinguish between these two cases. The Crab filaments serve as the nearby laboratories for understanding the very much larger filamentary structures that have formed in the intracluster medium of cool-core galaxy clusters.