I have a roll of 100 feet of old-fashioned twisted pair telephone cable. I measured the impedance at one end with an impedance analyzer. The analyzer sweeps frequency from 10 kHz to 5 mHz; the frequency (horizontal) scale is linear rather than the more common logarithmic format. The measured impedance is shown on a vertical scale ranging from 10 ohms at the bottom to 1000 ohms at the top.
Here is the impedance with the other end of the cable open circuited:
Here is the impedance with the other end of the cable short circuited:
Here are the previous two curves superimposed. Where they cross is the characteristic impedance of the cable. There is a marker at that frequency, and in the upper right of the image can be seen the value of the impedance, which is 111 ohms.
With a termination of 50 ohms, we get this impedance sweep:
Now with terminations of 50 and 100 ohms superimposed:
We can see that the 100 ohm termination results in an almost flat measured impedance. This means that the Zo is about 100 ohms. The earlier curves that intersected at 111 ohms gave a more accurate Zo.
These curves only show the magnitude of the measured Z; the true measured Z has a slight non-zero phase, but these curves are very close to the true Zo.
At any particular single frequency, the Zo of the cable can be found like this:
Measure the impedance at one end with the other end open; call this Zopen. Measure the impedance at one end with the other end shorted; call this Zshort. Zo is given by SQRT(Zopen*Zshort).
On the third image, at any frequency Zo is halfway between the curve for an open and the curve for a short. Of course, where the curves for open and short intersect shows Zo directly.