In an effort to search for Ly$alpha$ emission from circum- and intergalactic gas on scales of hundreds of kpc around $zsim3$ quasars, and thus characterise the physical properties of the gas in emission, we have initiated an extensive fast-survey with the Multi Unit Spectroscopic Explorer (MUSE): Quasar Snapshot Observations with MUse: Search for Extended Ultraviolet eMission (QSO MUSEUM). In this work, we report the discovery of an enormous Ly$alpha$ nebula (ELAN) around the quasar SDSS~J102009.99+104002.7 at $z=3.164$, which we followed-up with deeper MUSE observations. This ELAN spans $sim297$ projected kpc, has an average Ly$alpha$ surface brightness ${rm SB}_{rm Lyalpha}sim 6.04times10^{-18}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ (within the $2sigma$ isophote), and is associated with an additional four, previously unknown embedded sources: two Ly$alpha$ emitters and two faint active galactic nuclei (one Type-1 and one Type-2 quasar). By mapping at high significance the line-of-sight velocity in the entirety of the observed structure, we unveiled a large-scale coherent rotation-like pattern spanning $sim300$ km s$^{-1}$ with a velocity dispersion of $<270$ km s$^{-1}$, which we interpret as a signature of the inspiraling accretion of substructures within the quasars host halo. Future multiwavelength data will complement our MUSE observations, and are definitely needed to fully characterise such a complex system. None the less, our observations reveal the potential of new sensitive integral-field spectrographs to characterise the dynamical state of diffuse gas on large scales in the young Universe, and thereby witness the assembly of galaxies.