High-redshift binary quasars provide key insights into mergers and quasar activity, and are useful tools for probing the spatial kinematics and chemistry of galaxies along the line-of-sight. However, only three sub-10-kpc binaries have been confirmed above z=1. Gravitational lensing would provide a way to easily resolve such binaries, study them in higher resolution, and provide more sightlines, though the required alignment with a massive foreground galaxy is rare. Through image deconvolution of StanCam Nordic Optical Telescope (NOT) monitoring data, we reveal two further point sources in the known, z~2.38, quadruply lensed quasar (quad), J1721+8842. An ALFOSC/NOT long-slit spectrum shows that the brighter of these two sources is a quasar with z=2.369+/-0.007 based on the CIII] line, while the CIII] redshift of the quad is z=2.364+/-0.003. Lens modelling using point source positions rules out a single source model, favouring an isothermal lens mass profile with two quasar sources separated by ~6.0kpc (0.73") in projection. Given the resolving ability from lensing and current lensed quasar statistics, this discovery suggests a large population of undiscovered, unlensed sub-10-kpc binaries. We also analyse spectra of two images of the quad, showing narrow Ly{alpha} emission within the trough of a proximate damped Lyman-{alpha} absorber (PDLA). An apparent mismatch between the continuum and narrow line flux ratios provides a new potential tool for simultaneously studying microlensing and the quasar host galaxy. Signs of the PDLA are also seen in the second source, however a deeper spectrum is still required to confirm this. Thanks to the multiple lines-of-sight from lensing and two quasar sources, this system offers simultaneous sub-parsec and kpc-scale probes of a PDLA.