Hi Daniel,
You should be able to extract a raster using the Extract by Mask tool. You are accepting the default cell size which is probably resulting in a large raster dataset.
I understand you have specified the Processing Extent Geoprocessing Environment while running the tool. Can you also please try to specify a larger Cell Size in the environment to see if the tool runs faster and create a smaller output? Please specify both the Processing extent and the Cell Size.
If the tool extracts the raster correctly, you can calculate the size of the raster for the larger cell size by multiplying the number of rows with the number of columns. Once you know the number of rows and columns for the larger cell size, you can also estimate the number of rows, columns, and size of the raster as well as the download time for the default or smaller cell size.
Please let us know if you still notice a discrepancy.
Thanks
Noman
Nawajish Noman, Esri