Calculating spatial width from power spectrum HWHM
Say I have a 1D (spatial) signal (resolution = $1000$) which is zero everywhere except from $x=250$ to $750$, where it equals one.


This is not "resolution". Resolution is 300 Dots Per Inch. In which case, we could say that the total physical length of your pulse is $\frac{500}{300} \approx 1.666$ inches (or any other unit of length).


  I ultimately want to calculate the spatial width of this signal using FFTs.


Why?


  I am dealing with a signal that evolves with time and wish to calculate the average "pulse" width over all the time frames, so I do not know the widths. 


If there will be multiple pulses of different widths on the same signal, then by opting to detect them with the FFT you are setting yourself a very big challenge because the FFT would return to you information about the signal as a whole. So you could, for example, derive an average rate of pulses (even using the algorithm that is presented here) but not the widths of individual pulses.

If it is somehow guaranteed that within a window of 1000 samples, there will be a pulse whose length is guaranteed to be staying well below 1000 samples and all we have to do now is detect where the pulse is and how long it is, then opting for the FFT is an overkill.

The usual way to detect pulse widths is via the simple use of a threshold and a counter. Once the signal's amplitude goes above the threshold, the counter starts counting and it stops once the signal's amplitude goes below the threshold. If you are going to operate in a noisy environment, then there are a number of improvements to that such as adding hysterisis to the threshold, so that it doesn't respond to very short "bounces" of the waveform and adaptive thresholding where the threshold limit would be derived from the given window of observation (here, from the 1000 samples).

If you absolutely have to work in the frequency domain, it might be better to look into the discrete wavelet transform (DWT), whose output is a time/scale(frequency) representation. But the actual detection of the pulse width is likely to be happening (again) using some form of threhsolding on the output of the DWT. (So, again, huge overkill).