300KB is not really very large, but to see where the space is going, you can post your JSON or look through it yourself using a text editor. A lot of things can affect the JSON size. One of the worst is to use deform keys on a mesh with vertices that are affected by many bones. In that case, Prune can be used to reduce the number of bones that affect the vertices.
Weights - Spine User Guide: Prune
An even better way is to prefer using weighted bones rather than deform keys.
Note that using JSON is never going to be small. It is a text format meant to be readable by humans. For a smaller size and faster loading, you should use the binary skeleton format. You can also compress (eg deflate/zip) the binary data to get it a little smaller.
