I could relatively easily take the feeder down and use a tape measure but it’s more of a challenge to learn how to measure the length using my oscilloscope -Red Pitaya - but I have to admit I have no idea how to go about it.
A quarter wavelength of coax (shortened by its velocity factor) is a notch filter if the far end is left open.
So leave the far end open; and try feeding the coax from a frequency generator voltage source plus a series resistor at the near end; and sweep the frequency across a slightly wider range than is appropriate for an adjusted quarter wavelength to be your min to max estimate of your coax length. Use two scope probes on points before and after the series resistor, and look for the frequency causing the maximum drop in the ratio of waveform levels between the two scope inputs.
Well this is rather simple and in reality you dont need to ensure the far end is open, you can leave your antenna attached and it will still work just fine if all you are trying to do is determine the length of your feedline.
Before I get into the explanation let me share a video I did years ago demonstrating the exact setup you are trying to do using a function generator and oscilloscope to determine the length of a feedline. You can see that video here.
So its actually really simple, you hook a line from the output of your function generator into a Tee connector which connects to the port of your oscilloscope and the other port into the end of your feedline that is in your shack. Next you setup your function generator to produce a pulse output, that is a the a short pulse repeatedly over and over, make sure the pulse is significantly shorter in time than the space between pulses. While not strictly needed you will get the best results if yoru function generators output impedance is set to match the feedline (usually 50 ohms) and if your oscilloscope is set to high impedance mode (the default and sometimes only mode for some lower end o-scopes).
Next you set the trigger on your oscope until its able to detect the pulse. At this point what youll notice is the initial pulse on your oscope created from the outgoing pulse, but then shortly after that shifted a bit to the right you will se a second pulse. The second pulse may be of a slightly different shape or may even be completely inverted depending on the characteristics of your antenna. Now since you only care about the length of your feedline the actual shape of the return pulse really doesnt matter. What you want to do is compare the time difference between the very start of the rise on the first pulse with that of the rise (or if inverted fall) of the second pulse. Not that time difference and we will call that Δt.
So Δt is basically the time it took for the pulse to go from the feed point on your feedline where the o-scope is connected to your antenna and back again. Therefore since we only care about the length of the antenna we really only care about half that value, which will tell you the time it took for the pulse to go from your oscope to the antenna.
Now electrical waves through a feedline move fairly close to the speed of light but still not quite. Look up the model of feedline you have and in the datasheet a value is always reported called the Velocity Factor, this represents the percentage of the speed of light a wave will move through your feedline so basically c = C*vf, where c is the speed of an electrical wave in your feedline, C is the speed of light in a vacuum, and vf is the velocity factor of your feedline.
So now we know the speed the wave moves, and the time it takes to traverse the feedline, so it is a simple matter to actually calculate the distance, that would just be c*Δt/2, and now you have your feedline length.