Why doesn't a nuclear fuel pool become irradiated? Why doesn't the volume of water in a nuclear fuel pool become irradiated?  Why wouldn't the water around the pool become radioactive and circulate around making the whole thing deadly?
My question spawned from this cartoon from XKCD. 
The Spent Fuel Pool
 A: There aren't many radioactive isotopes of Oxygen and Hydrogen and the ones that there are aren't very radioactive.
As dmckee notes, there is Deuterium formed from Hydrogen capturing a neutron, this produces D$_2$O, or heavy water. But Deuterium is stable and so doesn't cause radioactivity in itself. Heavy water is chemically a little toxic but not a radiation risk. You could produce Tritium when Deuterium captures another neutron, but the rates of this happening at fuel rod energies/intensities is tiny.
The most likely source of fuel pond becoming dangerously radioactive is a crack in one of the fuel rods allowing isotopes generated in the fuel to leak out.
A: There a few more subtleties to this problem.  While only a "small" fraction of neutron absorptions go to something that will be a dangerous activation product, this doesn't assure us of safety.  That's because the flux coming from the fuel rods is many orders of magnitude higher than our safety limits to begin with.
In fact, consider the case of boiling water reactor (BWR) primary radioactivity concern from the coolant during operation.
http://en.wikipedia.org/wiki/Boiling_water_reactor#Disadvantages

shielding and access control around the steam turbine are required during normal operations due to the radiation levels arising from the steam entering directly from the reactor core. This is a moderately minor concern, as most of the radiation flux is due to Nitrogen-16, which has a half-life measured in seconds, allowing the turbine chamber to be entered into within minutes of shutdown.

This is a good reference point, because the spent fuel pool is qualitatively an extremely similar system.  In a BWR, the coolant passes directly through the nuclear core at full power.  The neutron flux in that environment is sufficiently high to activate the coolant to a dangerous level.  The coolant is then removed from the core, powered by a pressure difference maintained actively through feedwater pumps.
Also note that N-16 is an unavoidable product of exposing water to a neutron flux.  The reaction is the following:
$$ { }^{16}_{8}O + { }^{1}_{0} n \rightarrow { }^{16}_{7} N + {}^1_1 p$$
Almost all natural Oxygen is the O-16 isotope.  So if you're using water in a neutron flux, you'll have this.  So the question comes down to "how much" and "where".
Radioactivity in spent fuel pools is almost entirely from the most recent fuel because activity declines over time due to nuclear decay.  As you shut off the reactor, you go down to about 7% power immediately, relative to its full power.  There are regulatory minimum hold times in the core before you start shuffling around the fuel, and its power output declines further during this time.  After a few days, the power will certainly be below about 1% of full power.
This isn't very comforting, but that's because I've left out a critical detail - the major of the heat in spent fuel comes from decay transitions that do not emit neutrons.  While the heat goes down by these tangible factors, the neutron flux declines relative to the full power core by a much larger factor.
Those decays still give off copious gamma rays.  In fact, I believe that's the main hazard.  However, these are shielded very well by water, and unlike neutrons, they don't contribute significantly to activating constituents in the water.
For the N-16 that is produced in the spent fuel pool, we also have a critical difference that it operates by natural circulation.  After a few minutes, the N-16 has almost completely decayed away.  If the circulation flow takes this long for the water to circulate, then it can't be a hazard, but it's not clear that this is always the case.  The impurities in the water which can be activated more readily are also a concern.  But the problem comes back to the neutron flux to begin with, and it's so low that activation products won't be much of a concern.  The proportion of delayed neutrons is handled quantitatively in reactor point kinetics, and it is a very tiny fraction - vastly lower than decay heat itself.  It also declines over time.  Those delayed neutrons are actually critical to the control of the reactor, but constitute very low flux values when considering days or weeks after shutdown.
