Would 10 metres of liquid air be as effective against cosmic radiation as Earth's atmosphere? Assuming ballpark figures which give a depth of 10m if the Earth's atmosphere was liquefied, would that be as effective a protection against ionizing radiation from space as the gaseous atmosphere is?
 A: For many—but not all—shielding processes the parameters of interest are proportional to the areal density, 
$$ \text{Range} \propto \frac{\text{areal density}}{\text{mass density}} = \frac{\sigma}{\rho} \,,$$
so a first expectation would be that the same areal density of material (of roughly the same composition) will have the same effect. I don't have the density of liquid air on hand, but if it is near that of water then your 10 meters is about the same areal density.
The first exception that comes to mind is cosmic muons: their range in the atmosphere is controlled largely by their lifetime and a non-trivial number decay before reaching sea level (this is the biggest component of the altitude dependence of the cosmic background dose). Your liquid air buffer will be no help there.
In the same vein there may be an extra component from the K-long products of primary and secondary interactions. Their $10^{-8}\,\mathrm{s}$ lifetime is not an issue with kilometers of atmosphere above you, but could be with only 10 meters of shielding. That's probably it for neutral survivers: the strange neutral baryons have rather shorter lifetimes (a $\Lambda^0$ lasts on average $3 \times 10^{-10}\,\mathrm{s}$ or about 10 centimeters times the Lorentz factor; few will have $\gamma > 100$).
A second issue where length rather than matter comes into play is with the consequences of neutron spallation. Once thermalized (which is areal density dependent) the neutrons spread out on a random walk with step size also inversely dependent on the areal density. But the time they spend walking depends on the neutron's lifetime, so the number of steps is larger for denser materials leading to a range falling slower than inversely in density
$$ \text{Range} \propto \sqrt{\frac{\tau}{v_\text{thermal}\, \rho}} \,.$$
So a larger fraction of spallation neutrons make it into your inhabited volume.
