What electric field strength in the range 50MHz to 3.5GHz should be considered "natural"?
Black body radiation may be the thing you are looking for: -



The graph above covers above visible light wavelengths down to wavelengths of 100 um (3,000 GHz). The formula is called Planck's equation: -



where B is the spectral radiance, T is the absolute temperature of the black body, kB is the Boltzmann constant, h is the Planck constant, and c is the speed of light.

This also applies to more mundane frequencies but of course the powers are (thankfully) much smaller. The earth is the "black body" and you can assume it to be largely at 273 kelvin.

You could use this calculator to get the emissive power at any wavelength. Or maybe just make a spreadsheet but the great thing about the caluculator above is that it calculates the total emission across a given bandwidth (that you enter).

You then need to convert it to watts per square metre in order to determine the E and H field quantities because, E and H multiplied together produce watts per square metre. 

As per normal radio waves, E is 377 (120\$\pi\$) times bigger than H because 377 ohms is the impedance of free space.

Finally, if you are going to check this value with an antenna and some very sensitive electronics cooled down substantially you will need to take into account the antenna's aperture (defined in square metres).

Further reading on black body radiation related to antenna theory.