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Okay, so assuming the person is in the middle of nowhere, naked, and that 3K is the blackbody temperature of the surrounding universe, with some simplifications one could apply the following equation:
Q = S*A*E_s(T_b^4 - T_s^4)
Where
S is the Stephan Boltzmann constant, 5.67x10^-8 W/m^2-K^4
A is the surface area of the person's naked body, 1.8m^2, roughly
E_s is the emmisivity of human skin, 0.97
T_b is the temperature of the body in K, here ~310K (I have assumed that the body temp is uniform)
T_s is the temperature of space in K, 3K
Q is about 900 Watts. The typical metabolic heat generation for a sleeping person with 1.8m^s worth of skin is 40W/m^2 is 108 W. This means that your net heat loss would be ~800 W.
Assuming the the specific heat of the human body is approximately the same as the specific heat of water, (C_p = 4187 J/kg-K), a 200 lb person (mass = 90 kg), and that we're only concerned with temperatures above 0 C, you could figure out how fast your temperature would be dropping using the following equation:
dT/dt = Q/(mass*C_p)
dT/dt is about 2 thousandths of a degree Celcius per second. So if your body was uniformly 310K, initially you'd be losing a degree every 8 minutes. It would take about 40 minutes of naked stillness to reach stage 3 hypothermia.
There are probably errors in this analysis. And even if error free, it is pretty approximate. If you were behind the earth you'd be getting some heat from the earth, so you'd take a little longer to freeze to death.
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