What is the soil type you are working in? I think your plan sounds good - you are suggesting doing a cut-and-cover method with drainage "to daylight" which is great. When building things underground, water intrusion is an issue. If you are at the toe of a slope, there will be more water than if you were higher up. How much area there is upslope, how much rainfall you get, how fast your soil drains, and how deep the water table is will all be important to know.
I am not a structural engineer of any sort, but I would be wary of building my walls out of field stone and clay. A buried structure will have to carry the load over it on the roof, but there can also be "squeezing" forces that will try and buckle the walls at the invert (floor).
You will not be deep enough to develop a zone of arching, but imagine all the dirt on either side trying to slide down a wedge and into your excavation. How much pressure will there be? Without an engineer, youd only be able to guess. Some soils are very cohesive, like heavy clay, so they would resist lateral movement. Some soils are loose, and would be hazardous to even dig in. So while clay will hold its shape, it also does not drain quickly. If water can build up on the back side of your walls, because the backfilled soil is less dense for example, then you will get hydrostatic pressure that will try and push the walls in at the bottom. If you have ever seen concrete bow out the bottom of a form, you will know that it does not take a whole lot of depth to create some very large loads.
I would suggest you make the walls stiff. Concrete block with rebar-reinforced pillars within some or all of the cells would likely be plenty. Also, put some drainage at the base of the walls on the outside, and backfill with something permeable.