For steady flow, RAS will report the cumulative volume in a reach. From the Profile Output Table go to Options and select define table and look for volume.
So if you modeled the reservoir as a series of large cross sections you could back out the volume.
If you want a really accurate answer, create a 2D model of the reservoir.
You can specify the starting water surface in the 2D area. Create a short dummy run and then go look in the computation log file (from the Unsteady Run Editor, Options), it will show the starting water volume in the 2D area.
I guess you could also start with it dry and just run a flow boundary until it fills up and compute the volume from the flow.