If the reels are yours, track how many feet or meters you have used. If you have not been doing this, you could use a scale to ESTIMATE what's left. However, you need to know what is the mass/meter and the mass of the case [which varies 1-3 grams]. You also need a food scale that measures 0.1 grams. Ounces are useless.
Somewhat related, I once sold two brand new reels of string on eb@y, and the guy was pissed off claiming I used half a set from one of the reels. So in disbelief I asked by what possible means could he have calculated that out, while also offering to refund him whatever that half a set would be valued. The dude was incredulous, asking how dare I doubt him, and that he had his ways. Uh, how dare I doubt him?! He was the one throwing around accusations. He refused the partial refund and said that he'll just never do business with me again...which was doing me a favor; wouldn't want to have to deal with his nonsense again. I'm assuming he just weighed both reels and noticed one weighed a half set's worth of string more than the other.
I make it a point to weigh the empty plastic reels when I finish one and I try to remember to check the weights before and after using a couple racquets worth of string so I have a reasonable idea of wgt/ft. I enter this info in table which is set up to calculate what the reel should weigh when full: core + (Wgt/ft *reel length). I check reels when they arrive. They do vary somewhat. Some reels I feel I get shorted a bit. Others I get a little extra. I think it evens out over time.
Here's what my table currently look like. As you can see I am missing a bunch of data but as a general rule, tennis string ends up weighing about .5gms/ft.