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We present 850$μ$m polarization observations of the L1689 molecular cloud, part of the nearby Ophiuchus molecular cloud complex, taken with the POL-2 polarimeter on the James Clerk Maxwell Telescope (JCMT). We observe three regions of L1689: the clump L1689N which houses the IRAS 16293-2422 protostellar system, the starless clump SMM-16, and the starless core L1689B. We use the Davis-Chandrasekhar-Fermi method to estimate plane-of-sky field strengths of $366\pm 55$ $μ$G in L1689N, $284\pm 34$ $μ$G in SMM-16, and $72\pm 33$ $μ$G in L1689B, for our fiducial value of dust opacity. These values indicate that all three regions are likely to be magnetically trans-critical with sub-Alfvénic turbulence. In all three regions, the inferred mean magnetic field direction is approximately perpendicular to the local filament direction identified in $Herschel$ Space Telescope observations. The core-scale field morphologies for L1689N and L1689B are consistent with the cloud-scale field morphology measured by the $Planck$ Space Observatory, suggesting that material can flow freely from large to small scales for these sources. Based on these magnetic field measurements, we posit that accretion from the cloud onto L1689N and L1689B may be magnetically regulated. However, in SMM-16, the clump-scale field is nearly perpendicular to the field seen on cloud scales by $Planck$, suggesting that it may be unable to efficiently accrete further material from its surroundings.