How data hungry is it, or what is the minimum volume of data needed before its worth investigating?

The more complex the data is, the more you need. If your values are always 5, then you need only one data point.

If your values were always 5,you wouldn't use an LSTM to model it either. So presumably there's a threshold for when LSTM becomes practical and useful, no?

Sure, that was an extreme example. The point was that the minimum of data is 1 point, maximum is "all of it". It entirely depends on your use case.