Also, in case an environment with equal density of air and water can be made:
Will it be possible for a human to breathe in that environment?
[Physics] Can we make an environment where the density of air equals the density of water
densityfluid dynamics
Related Solutions
This is an instrument that measures fog density and has an experimental plot, figure 9 .
Once you have the relative humidity at the fog appearance at a temperature and pressure , one can use known equations to get the density. This link gives a calculator.
For a complete answer, see this identical question on movies stack exchange. However, the brief answer is: P-Traps and vent pipes.
If the water from the room were to make the toilet act as a drain, there would still be air from the vent pipe coming from above, as the water would flow downwards. It would be a nightmare trying to get the tube into the correct position, but possible.
Best Answer
One way to discuss the maximum density of a gas is to look at its phase diagram. In a PT phase diagram, we will obviously seek a point on the boundary of the gas region where the temperature is low and the pressure is high. For at least the Oxygen and Nitrogen phase diagrams, it looks like the critical point would be a good candidate for these maximum gas density conditions. Thus, identifying the density of air at its critical point would be a good possibility to get us its maximum density or close to its maximum density.
The critical point for air has a specific volume of $92.35 cm^3/mol$, and a density of $0.31 g/cm^3$ (by using its average formula mass). This is less than the density of water, which is $1.0 g/cm^3$ by definition.
Because of this, there seems to be a good case that the answer to your question is no. If the conditions are low enough temperature and high enough pressure to make the substance of air the same density of water (if possible at all), it will no longer be a gas, and most people's layman concept of "air" requires that it is a gas. To answer your second part, the critical point of air is at $-221^{\circ} F$ and the pressure is $37.25 atm$, so no, a human could not withstand these conditions.
Update:
I was unaware of how useful Wolfram Alpha has become. Here is air at conditions sufficient to produce a density greater than water:
http://www.wolframalpha.com/input/?i=air+at+2500+bar+and+-130+degrees+C
Here is another example, at $0^{\circ} C$:
http://www.wolframalpha.com/input/?i=air+at+5000+bar+and+0+degrees+C
(Image copyright Wolfram alpha, use believed consistent with terms)
Both of these lie squarely in the supercritical liquid region. Is that a gas, no, but it's not fully a liquid either. I suppose the next question is if you can breathe such a fluid. Well, for these conditions we're talking about 2,500 and 5,000 times atmosphere pressure. The deepest trench in the ocean has a pressure of 1,000 times atmosphere at the bottom (and no human has been there). Super cold temperatures are completely intolerable for humans, but pressure can be increased dramatically. In order to have breathable air the same density of water, we would have to create pressures many orders of magnitude times the world record of what has been withstood by a human. Whether or not it is absolutely impossible or not is kind of our of our scope. All I can say is that it seems unlikely.