Is it possible in SQL to run a query of spatial data, and then just return a single polygon, or a radius, that encloses, aggregates, all of the spatial data that was in that query into one feature?
Yes, using SQL Server's
You essentially have three options
ConvexHull: returns a polygon that essentially wraps the input with a rubber band,
EnvelopeAggregate: returns a bounding box (polygon) that encapsulates the input (box = lines in a cartesian project, arc segments in a 3d/spheroid/geographic projection)
UnionAggregate: takes all of the inputs and composes them into a single complex-polygon
Note: you mention radius, a radius is a buffer over an input in GIS.
If you need more features, you should check out the free PostgreSQL and PostGIS which supports even more GIS functionality.