How big is the area? Consider the Black Sea basin (or Mediterranean). These two areas have been intermittently blocked off from the ocean over geologic history. Evaporation will then remove the water at some rate. As long as the rate of evaporation is greater than the rate of incoming water then the basin will remain dry.
You can have as many rivers entering the basin as you wish as long as they don't bring in more water than can evaporate. The interior will likely be very dry due to the rain shadow effect from the mountains, but the mountains will capture that moisture and convert it into rain in the mountains itself. The outside of the ring of mountains is likely to be fairly wet and might be where some tribes will prefer to live.
How far below sea level the interior end up being will influence temperature. Temperature gets hotter the lower in the atmosphere you get. The bottom of the empty Mediterranean basin, for example, was likely 20 degree hotter than the current sea level. This increased temperature would probably have resulted in a particularly nasty desert broken only by ribbons of green around the rivers falling into the basin and with a deadly hypersaline lake in the center.