You were right to question whether the atmosphere really held that much water. It comes nowhere close! We use precipitable water to track this, which is the measure of all moisture in the entire column of air in the troposphere. We can get good widespread estimates from remote observations. This animation shows current amounts of precipitable water levels across the US from satellite. Here is a still from this afternoon (Sunday, August 27) during the Harvey event:
75 mm is less than 3 inches.
We also get in-situ exact measurements worldwide from twice daily radiosonde balloon launches. Unfortunately none are located right near the locations receiving excess rainfall. But you can check US sites of current precipitable water measurements any time by going to SPC's sounding page and looking for the PW value in the left side of the bottom table. SPC also maintains a climatology page for precipitable water and other values. You can see there that 3 inches (76 mm) of moisture in the sounding is extremely rare, and no US site has ever had 4 inches (102 mm) of precipitable water.
However, the answer is also not truly found in evaporation rates. The conditions in strong hurricanes do greatly enhance evaporation rates due to the high wind speeds and warmer waters. Measurements are actually a bit difficult to come by in such extreme conditions, with challenges in isolating evaporation effects from spray as well as in getting the instruments positioned into such environments (new field campaign: who wants to take the research ship out into the category 5 hurricane!?!). As this 2007 study by Trenberth et al. noted:
We are unaware of reliable estimates of evaporation in hurricanes, and published measurements do not exist in winds above about 20 m s−1 although some progress has been made in the Coupled Boundary Layer Air-Sea Transfer Experiment (CBLAST)
However, in that study model analyses suggested that evaporation rates in the core of hurricanes are likely no more than 1-2 inches (25–50 mm) per day.
How is that possible?
It's quite important to notice that heavy rainfall - well over 5 inches (125 mm) - can actually fall in as little as a couple hours almost anywhere, such as in this 2015 flood event in Nebraska. How can that be?
The secret lies in the nature of even the weakest storm: even as a storm is just beginning to form, it begins to draw in air from surrounding areas. This is NOAA's diagram of a typical developing thunderstorm cell:
You can see the curve at the arrows near the bottom, indicating inflow of surrounding air into the storm. This inflow turns a thunderstorm into more of an engine, processing a continuing stream of incoming air, removing its moisture. In a single cell thunderstorms in an environment without background winds, the "waste" air will eventually pile up and choke off the influx. But even in such circumstances, a few inches of rain may fall. That isn't by using up all of the moisture from the cloud's environment, but instead by using just a portion of the moisture from the reservoir in and around the cloud.
If some upper level winds exist to help exhaust the "spent" air, storm systems can persist for even longer periods of time. For example, the Midwestern United States commonly sees long-duration late summer heavy rainfall initiated along stationary boundaries in which inflow lacks direct access to significant warm water bodies.
Larger systems that dump huge amount of precipitation over greater areas must pull in a more consistent, stronger inflow of warm, moist air from greater distance. Examples of this happening include the Pineapple Express for rainfall in California/the southwest US, the low-level jet for spring and summer squall lines in the Plains, onshore winds during the Indian monsoon, and air from off the Gulf Stream in Nor'easters.
In all these regional large precipitation events moist air gathered from a great surface area flows into a smaller region. As the air approaches, it is lifted by the low pressure and its associated features, condenses, and finally falls as rain (or snow). This process is often termed "moisture convergence". The Storm Prediction Center also offers plots of localized deep moisture convergence [choose a region, then look under the upper air menu]. The convergence contours, shown in red, really show the piling up of humidity that is causing the heavy rainfall in Harvey:
But perhaps to visualize the scale involved in creating a catastrophic largescale flood such as Harvey, this plot, created from a base image from pivotalweather.com, best shows the conditions around the storm (from the GFS model):
Basically the atmosphere of the entire Gulf (and beyond) is being pumped into the southeast Texas area. So although the air can only hold a couple inches (some 50 mm) of water, and evaporation rates are typically only a fraction of an inch (several mm) per day... bringing that together from such a large source region, and focusing it down into one small area... can lead to these awful extreme deluges.
Addendum: It should also be highlighted that the NHC adds in their report on Harvey
that rising motion was also enhanced by a front which had stalled in the area. Air being advected in by Harvey's flow would naturally rise over that layer of cooler air when moving inland (a process called isentropic lift), which proves particularly efficient in condensing out the (abundant) moisture en masse into rainfall. Most substantial regional floods require similar existence of a significant broad lifting mechanisms overlaid with such a relentless inflow of warm, moist air.