You can't measure the characteristic impedance without a TDR.
You don't really need to, either. Since you're DIYing, you can either:
a) Use 300 ohms for wire that has 300 ohms on the box;
or,
Be anal and make the cable with 270, 300, 330 ohms, pick the best-sounding one, then bracket +/- 30 ohms to be sure, then bracket +/- 10 ohms, repeat centered on the new value, bracket +/-5 ohms, repeat, bracket +/- 1 ohm, repeat. But you'd have to make 24 cables, or modify the original pair 23 times -- so I'd at least suggest stopping when you can't hear a difference! In all fairness, though, you might not hear a difference at even the first step. The advantage of the bracketing procedure is that you'll find the best value compromise between the termination of your cable and the load on your system, which could be many many ohms off the actual impedance of the cable. You might therefore end up with a cable that sounds even more mindblowingly-awesome than a "proper" ProSink-ed cable, but that sounds awful when you put it on a different rig and/or change your source and/or amp.
The idea of bracketing and the idea of the best value for the termination resistor being different than the impedance of the cable have not been tested (by me, anyway). Use at your own risk (of sanity, in any case).