EX Lup is the prototype of the EXor class of young eruptive stars: objects showing repetitive brightenings due to increased accretion from the circumstellar disk to the star. In this paper, we report on medium-resolution near-infrared spectroscopy of EX\,Lup taken during its extreme outburst in 2008, as well as numerical modeling with the aim of determining the physical conditions around the star. We detect emission lines from atomic hydrogen, helium, and metals, as well as first overtone bandhead emission from carbon monoxide. Our results indicate that the emission lines are originating from gas located in a dust-free region within ~ 0.2 AU of the star. The profile of the CO bandhead indicates that the CO gas has a temperature of 2500 K, and is located in the inner edge of the disk or in the outer parts of funnel flows. The atomic metals are probably co-located with the CO. Some metallic lines are fluorescently excited, suggesting direct exposure to ultraviolet photons. The Brackett series indicates emission from hot (10000 K) and optically thin gas. The hydrogen lines display a strong spectro-astrometric signal, suggesting that the hydrogen emission is probably not coming from an equatorial boundary layer; a funnel flow or disk wind origin is more likely. This picture is broadly consistent with the standard magnetospheric accretion model usually assumed for normally accreting T Tauri stars. Our results also set constraints on the eruption mechanism, supporting a model where material piles up around the corotation radius and episodically falls onto the star. ; Comment: 12 pages, 8 figures, 1 table, accepted for publication in ApJ