How can I build a subpanel off a 3 prong 220V dryer outlet?
Depends how much work you want to do. The simplest approach given: " I don't expect any more than a couple hundred watts of intermittent load per circuit."

Is to convert it to a single 120V circuit at 20 amps, 2400W capacity; (or 15A, depending what the total expected wattage actually is) that happens to have vastly oversized wires. Replace the dual-pole 30A breaker with a single pole 20A breaker, use one of the hots (retagged green) to ground, [or retagged white to neutral if the current "neutral" wire is already at the main panel ground/neutral buss and it's more convenient to make THAT one the "green/ground" wire] change the receptacle, perhaps add a few boxes connected with at least 12Ga copper to house some additional outlets, away you go.

Now, if it's easy to run a ground wire, (or by some miracle the cable actually has one already, unused) you can do that and convert to a 4-prong, and then my answer in the "near duplicate" linked above applies. Several other options were provided that ignored that poster's desire not to do anything in the panel - if you are OK doing things in the panel, they also apply if you like them better. You could replace the receptacle with a sub-panel, for instance.