Context. Inferences about dark matter, dark energy, and the missing baryons all depend on the accuracy of our model of large-scale structure evolution. In particular, with cosmological simulations in our model of the Universe, we trace the growth of structure, and visualize the build-up of bigger structures from smaller ones and of gaseous filaments connecting galaxy clusters.Aims. Here we aim to reveal the complexity of the large-scale structure assembly process in great detail and on scales from tens of kiloparsecs up to more than 10 Mpc with new sensitive large-scale observations from the latest generation of instruments. We also aim to compare our findings with expectations from our cosmological model.Methods. We used dedicated SRG/eROSITA performance verification (PV) X-ray, ASKAP/EMU Early Science radio, and DECam optical observations of a similar to 15 deg(2) region around the nearby interacting galaxy cluster system A3391/95 to study the warm-hot gas in cluster outskirts and filaments, the surrounding large-scale structure and its formation process, the morphological complexity in the inner parts of the clusters, and the (re-)acceleration of plasma. We also used complementary Sunyaev-Zeldovich (SZ) effect data from the Planck survey and custom-made Galactic total (neutral plus molecular) hydrogen column density maps based on the HI4PI and IRAS surveys. We relate the observations to expectations from cosmological hydrodynamic simulations from the Magneticum suite.Results. We trace the irregular morphology of warm and hot gas of the main clusters from their centers out to well beyond their characteristic radii, r(200). Between the two main cluster systems, we observe an emission bridge on large scale and with good spatial resolution. This bridge includes a known galaxy group but this can only partially explain the emission. Most gas in the bridge appears hot, but thanks to eROSITA's unique soft response and large field of view, we discover some tantalizing hints for warm, truly primordial filamentary gas connecting the clusters. Several matter clumps physically surrounding the system are detected. For the "Northern Clump," we provide evidence that it is falling towards A3391 from the X-ray hot gas morphology and radio lobe structure of its central AGN. Moreover, the shapes of these X-ray and radio structures appear to be formed by gas well beyond the virial radius, r(100), of A3391, thereby providing an indirect way of probing the gas in this elusive environment. Many of the extended sources in the field detected by eROSITA are also known clusters or new clusters in the background, including a known SZ cluster at redshift z = 1. We find roughly an order of magnitude more cluster candidates than the SPT and ACT surveys together in the same area. We discover an emission filament north of the virial radius of A3391 connecting to the Northern Clump. Furthermore, the absorption-corrected eROSITA surface brightness map shows that this emission filament extends south of A3395 and beyond an extended X-ray-emitting object (the "Little Southern Clump") towards another galaxy cluster, all at the same redshift. The total projected length of this continuous warm-hot emission filament is 15 Mpc, running almost 4 degrees across the entire eROSITA PV observation field. The Northern and Southern Filament are each detected at >4 sigma. The Planck SZ map additionally appears to support the presence of both new filaments. Furthermore, the DECam galaxy density map shows galaxy overdensities in the same regions. Overall, the new datasets provide impressive confirmation of the theoretically expected structure formation processes on the individual system level, including the surrounding warm-hot intergalactic medium distribution; the similarities of features found in a similar system in the Magneticum simulation are striking. Our spatially resolved findings show that baryons indeed reside in large-scale warm-hot gas filaments with a clumpy structure.