We investigate a high-capacity, space-division-multiplexed (SDM) transmission system using self-homodyne detection (SHD) in multi-core fiber (MCF). We first investigate SHD phase noise cancellation with both kHz and MHz range linewidths for both quadrature-phase shift-keyed (QPSK) and 16 quadrature-amplitude modulation (16QAM) signals, finding that phase noise cancellation in SHD enabled transmission with MHz linewidth lasers that resulted in error floors when using intradyne detection. We then demonstrate a high throughput SHD transmission system using low-cost, MHz linewidth distributed feedback lasers. We transmit a CW pilot-tone on a single core of a 10.1 km MCF span with the remaining 18 cores used to transmit 125 wavelength-division multiplexed (WDM) QPSK and polarization-division-multiplexed (PDM)-QPSK signals with 50 GHz channel spacing at 25 GBd. For PDM transmission and assuming a 7% forward-error correction overhead this is equivalent 210 Tb/s transmission with a SE of 33.4 b/s/Hz. High-capacity transmission is achieved despite high inter-core crosstalk, broad transmitter linewidth and narrow channel spacing, showing that combining SHD with MCF enables high throughput, low-cost transmission in next-generation optical networks.