You can calculate the NDWI using basic raster calculator in arcmap.
You need to use Landsat bands 3 and 5 (Landsat 8 OLI), and simply calculate a new raster with an algebraic expression:
(Band 3 - Band 5)/(Band 3 + Band 5)
You should preferably use images corrected to TOA.
In this example I have bands 3 and 5
The result is a raster with values between -1 and 1 (hence the "Normalized" in the index name. You can see in the histogram that water values are close to 1, and the rest of the pixels are between -1 and 0.
So to calculate the area, you first have to extract these values into a separate raster with a raster calculator or reclassify tool, the result of which you can convert into a vector layer, which you can then easily calculate its dimensions.