Waterdeep is approximately 3.8 miles x 1.5 miles
Or ~20,000 feet north/south and ~8,000 feet east/west. Both measurements were done along the greatest straight-line dimension of the city. This measurement is done by measuring the space between the walls of Waterdeep, so it includes the enclosed section of the harbor to the south.
This was derived from the map provided in the 3rd edition book City of Splendors: Waterdeep, which actually includes a scale. A copy of this map can be found here if you would like to make more precise measurements of distances between things.
I checked other sources that relate to Waterdeep, such as Volo's Guide to Waterdeep (2nd Edition), and did not find any contradicting maps. The 2nd Edition adventure titled Waterdeep includes a map, but no scale. The 1st edition sourcebook Waterdeep and the North does the same...map, but no scale.
Thus, with no contradicting evidence, it's probably a fair assumption that the 3rd edition scale of the city is accurate.