What you basically ask for can be done (I've done it), but not all your specific constraints can be reasonably met.
The way to measure transmission line impedance is to drive a step onto the transmission line with a known resistance, while observing the waveform on a scope. During the time the step propagates to the end of the cable and back, the cable will look like a resistor at its characteristic impedance.
The out and back time needs to be long enough for your scope to settle to an obvious level. You say you only have a 50 MHz scope. That means, in theory, even if you fed it a 50 MHz square wave, it would look like a sine. Each level of a 50 MHz square wave lasts 10 ns, which is the guaranteed to fail case for looking at a level with your scope. I'd want any level to be at least 10x that, so 100 ns.
Light propagates 30 m in 100 ns thru free space. A rough rule of thumb is that propagation in the kind of transmission line you will be measuring is about ½ the speed of light. That requires 15 m. Since you get the time for the signal to propagate to the end of the cable and back again, you only need that cable to be about 7.5 m in length. Let's say 10 m is a good minimum requirement for your setup. Traces on PC boards are out of your realm.
Other than needing about 10 m of cable, the rest is easily doable with your equipment. Set the function generator to a 100 kHz square wave. The amplitude isn't important as long as the square wave is clean and easily visible on your scope. Put 100 Ω or so resistance between the function generator output and one conductor of the cable. Tie the other cable conductor, the function generator ground, and the scope probe ground together at the entrance to the cable.
Trigger the scope on the rising edge, then look at the function generator output directly. Verify that it's a reasonably clean square wave and note its amplitude. Now look at the cable side of the resistor. You should see a reasonably clean step right after each function generator edge, but the signal will be attenuated.
The signal immediately after each edge is the result of a resistive voltage divider between the resistor you added and the cable characteristic impedance. Only very simple grade school arithmetic is needed to find the cable impedance given the series resistance and the attenuation factor.
For example, if the function generator is putting out a 5 V step each edge, you have a 100 Ω resistor in series, and the amplitude after the resistor is 3 V, then cable characteristic impedance is 150 Ω.
The best measurement is when you get a result ½ of the input. That means the cable resistance is equal to the deliberate series resistance. You loose resolution as the factor gets too close to 1 or 0. If it's outside the ⅓ - ⅔ range, change the resistor to bring it closer to ½.
But don't stop there. It's interesting to measure cable impedance, but there is a lot more you can see with this setup. You will see a reasonably flat level during the out and back propagation time, but pay attention to what happens after that. If the far end of the cable is left open, you will see the reflection cause steps of out-and-back length that eventually approach the unattenuated input voltage. If the far end is shorted, note that the initial step size doesn't change, but the subsequent steps approach 0.
What you are seeing is the result of reflections at the end of an unterminated cable. When you put a resistor of the characteristic impedance at the end of the cable, the reflections should theoretically stop. You will get some great intuition about cables, impedances, and reflections by experimenting with different driving resistances and termination resistances. Put a 200 Ω variable resistor at the end of the cable and watch the effects on the scope as the resistance is varied.
Then terminate the cable with the optimum resistance and put the variable resistor between the function generator and cable. You'll see different results when varying that.
See how sensitive reflections are to a little impedance mismatch.
Try to make the reflections go away as much as possible. See how close you can get. Then explain why the resistor at the function generator needs to be lower than you thought after determining the cable impedance.
The only problem is the length of the cable. We don't have so much cable at school
The cheapest cable for your purposes is probably simple unshielded twisted pair. A few 100 feet of that shouldn't cost all that much.
50 Ω or 75 Ω coax would be better in that the impedance will be more uniform and the signals cleaner. Maybe a local company is willing to donate a few 100 feet to the school. I'd ask the local phone and cable companies. They are usually big and often have formal programs to support communities. They might have 10 m pieces they toss out routinely.
Cat5 ethernet cable would work too. Even local electricians install that so will have some around. However, those are usually small enterprises, so they probably won't want to give you 100 m of cable they have to pay for. Large companies often have spools of Cat5 around. If you know someone at such a company, arranging a 100 m donation would probably not be too hard.