Indeed, you should use the MeasureImageAreaOccupied module, because it produces a measurement called “TotalImageArea”. In general in CellProfiler, you access measurements using DataTools > ExportData (or the equivalent module, ExportToExcel, built right into the pipeline). You will want to export the “Image” measurements.
Now, the only question is in what units do you want your measurement of image area to be? By default, CellProfiler assumes that your images are one micrometer per pixel (you will notice that the value in the “Pixel Size” box in the main CellProfiler window is, by default “1”). When this value is set to “1”, then any measurements that come out of CellProfiler are essentially in units of pixels.
If you want the measurements to be in units of micrometers, indeed, you will need to know the conversion factor (that is, the size each pixel represents as determined by a stage micrometer or from the manufacturer of the microscope). Enter this value the number of micrometers per pixel in the Pixel size box and any measurements produced will be in units of micrometers.
You can change the Pixel size for the current session by typing a different value in the Pixel size box, or more permanently using File > Set Preferences. You can read more about pixel size by clicking the “?” button next to the Pixel size box in the main CellProfiler window.