SRTM data has gaps mostly from terrain effects (radar shadows) - not [usually anyway] the atmosphere.
If you can identify the cells that are "gaps" by their value (poke around with the identify tool) then you can classify your tiff into two classes like this using the Raster Calculator tool:
Con("srtm_raster" > 1000, 0, 1)
You can then convert the raster to polygons and from there you can calculate areas. You may need to convert multipart polygons to single part (there's a tool for that) to get areas for each individual blob.
I'm assuming your tiff doesn't have NODATA cells. If the gaps are not crazy values but actually classified as NODATA, you would then do it this way:
Con(IsNull("srtm_raster"), 0, 1)