We have used Chandra to resolve the nearby 70 Oph (K0 V+K5 V) and 36 Oph (K1 V+K1 V) binary systems for the first time in X-rays. The LETG/HRC-S spectra of all four of these stars are presented and compared with an archival LETG spectrum of another moderately active K dwarf, Eri. Coronal densities are estimated from O VII line ratios and emission measure distributions are computed for all five of these stars. We see no substantial differences in coronal density or temperature among these stars, which is not surprising considering that they are all early K dwarfs with similar activity levels. However, we do see significant differences in coronal abundance patterns. Coronal abundance anomalies are generally associated with the first ionization potential (FIP) of the elements. On the Sun, low-FIP elements are enhanced in the corona relative to high-FIP elements, the so-called FIP effect. Different levels of FIP effect are seen for our stellar sample, ranging from 70 Oph A, which shows a prominent solar-like FIP effect, to 70 Oph B, which has no FIP bias at all or possibly even a weak inverse FIP effect. The strong abundance difference exhibited by the two 70 Oph stars is unexpected considering how similar these stars are in all other respects (spectral type, age, rotation period, X-ray flux). It will be difficult for any theoretical explanation for the FIP effect to explain how two stars so similar in all other respects can have coronae with different degrees of FIP bias. Finally, for the stars in our sample exhibiting a FIP effect, a curious difference from the solar version of the phenomenon is that the data seem to be more consistent with the high-FIP elements being depleted in the corona rather than with a low-FIP enhancement.