I am working with the Cost Distance tool and the numbers I am getting are the product of cost and distance (six/seven digit numbers, as my cell size is 5000 feet, my weights are 0-11). I have set up costs to be in minutes to cross a cell and want to make a map of my study area with distance to reach one of my sources. However, the cost distance output seems to be distances (very large numbers) not costs. If I run the Cost Path tool, and sum up the costs of the cells along my cost path with Zonal Statistics, I get the total I expect in minutes (in the case below 45.0 to travel 25 miles or so).

This page in the help seems to imply this can be done, note the picture of total costs to transport lumber.

Calculating cost of travel—ArcGIS Help | ArcGIS Desktop

Is there a way to do this in Spatial Analyst? I want to do this in raster because I want to model accessibility (in minutes) of all cells in my study area, including those that are away from roads.

Curtis... 'costs' are a relative term. In your case ascribing minutes to cross a cell is reasonable and allows you to model traversing open landscapes more readily. For your roads, you could translate speed limit to a travel time, hence modelling various road types. Similarly, landscapes like open field, hardwood forest etc can be ascribed a representative cost which would obviously be slower. I do this during the cost surface construction phase. For non-motor transit, path distance allows you to incorporate slope characteristics more representative of walking/hiking speed differences associated with terrain.

In your case, if you are travelling just along roads by vehicle, you can simply set your non-road cells to nodata and you will get similar results to a simple network analyst Orig-Dest study