Is it possible for a ground-based observer to calculate how much water is in a particular cloud?

Yes, if he or she has access to and is skilled in the use of a microwave radiometer. The cloud contains both water vapor and liquid water (in the form of tiny water droplets) OR ice crystals (when the cloud temperature is below freezing), which is sensitive enough to sense both. Ordinary radar can pick up precipitation density (when the droplets or crystals become heavy enough to fall as rain or snow (or hail)), but lack sensitivity to detect the droplets/crystals that make up the cloud. I assume by your question that you meant the water content of the cloud, and not how much rain or snow will fall from that cloud. A cloud always has water vapor and condensed water vapor (water droplets or ice crystals) in it, of varying amounts, but not all of it (often none of it)will fall as precipitation.

Thanks so much for your answer. You made it much easier for me to understand all of this now