Assuming you’re a reasonably good swimmer, you could probably survive
treading water anywhere from 10 to 40 hours. At that point, you would black out
from fatigue and drown. This is also true for a pool without nuclear fuel in
the bottom.
Spent fuel from nuclear reactors is highly radioactive. Water is good for
both radiation shielding and cooling, so fuel is stored at the bottom of pools
for a couple decades until it’s inert enough to be moved into dry casks. We
haven’t really agreed on where to put those dry casks yet. One of these days we
should probably figure that out.
Here’s the geometry of a typical fuel storage pool:

The heat wouldnt be a big problem. The water temperature in a fuel pool
can in theory go as high as 50°C, but in practice theyre generally between
25°C and 35°C—warmer than most pools but cooler than a hot tub.
For the kinds of radiation coming off spent nuclear fuel, every 7 centimeters
of water cuts the amount of radiation in half.
The most highly radioactive fuel rods are those recently removed from a
reactor. Based on the activity levels provided by Ontario Hydro in this
report, this would be the region of danger for fresh fuel rods:
Swimming to the bottom, touching your elbows to a fresh fuel canister, and
immediately swimming back up would probably be enough to kill you.
Yet outside the outer boundary, you could swim around as long as you
wanted—the dose from the core would be less than the normal background dose you
get walking around. In fact, as long as you were underwater, you would be
shielded from most of that normal background dose. You may actually receive a
lower dose of radiation treading water in a spent fuel pool than walking around
on the street.
That’s if everything goes as planned. If there’s corrosion in the spent
fuel rod casings, there may be some fission products in the water. They do a
pretty good job of keeping the water clean, and it wouldn’t hurt you to swim in
it, but it’s radioactive enough that it wouldn’t be legal to sell it as bottled
water. (Which is too bad—it’d make a hell of an energy drink).
We know spent fuel pools can be safe to swim in because they’re routinely
serviced by human divers.
However, these divers have to be careful.
On
August 31st, 2010, a diver was servicing the spent fuel pool at the Leibstadt
nuclear reactor in Switzerland.
He spotted an unidentified length of tubing on the bottom of the pool and
radioed his supervisor to ask what to do. He was told to put it in his tool
basket, which he did. Due to bubble noise in the pool, he didn’t hear his
radiation alarm.
When the tool basket was lifted from the water, the room’s radiation
alarms went off. The basket was dropped back in the water and the diver left
the pool. The diver’s dosimeter badges showed that he’d received a
higher-than-normal whole-body dose, and the dose in his right hand was
extremely high.
The object turned out to be protective tubing from a radiation monitor in
the reactor core, made highly radioactive by neutron flux. It had been
accidentally sheared off while a capsule was being closed in 2006. It sank to a
remote corner of the pool floor, where it sat unnoticed for four years.
The tubing was so radioactive that if he’d tucked it into a tool belt or
shoulder bag, where it sat close to his body, he could’ve been killed. As it
was, the water protected him, and only his hand—a body part more resistant to
radiation than the delicate internal organs—received a heavy dose.

So, as far as swimming safety goes, the bottom line is that you’d probably
be ok, as long as you didn’t dive to the bottom or pick up anything strange.
But just to be sure, I got in touch with a friend of mine who works at a
research reactor, and asked him what he thought would happen to you if you
tried to swim in their radiation containment pool.
“In our reactor?” He thought about it for a moment. “You’d die
pretty quickly, before reaching the water, from gunshot wounds.”
From http://what-if.xkcd.com/29/