Short answer: hugely variable, but generally very small.

Obviously some kings (Babylon, Assyria, Persia, etc.) referred to ruled over some extremely vast territories, with enormous populations. Outside of these superpowers, most of the kings in the Bible were fairly localized: they controlled a city and its immediate environs. Note that the (quite ahistorical) book of Joshua mostly assumes that each city has its own king.

The five cities of the Philistines, for instance, each had their own rulers (1 Sam 6:4) despite some political cohesion. (It should be noted that this kind of ruler is called a "seren" in Hebrew, using a loan-word rather than the usual "melek" for "King.")

Other kings are referred to by their lands, and we have a decent sense of their approximate territory, containing one to several significant cities, and many outlying villages. Aram in particular controlled a fairly large region, stretching at least from the capital Damascus (the kingdom is sometimes referred to as Aram-Damascus) to the borders of Israel.

But then comes the big question: what were the borders of Israel? How powerful really were the Israelite kings?

This is a tough question, particularly due to the political and religious implications it still carries. But the scholarly consensus is: not very powerful. Saul is a local enough ruler that he still personally plows a field (1 Sam 11), and his army wages war with a grand total of two swords (1 Sam 13). Ahab gets in a major dispute over a single vineyard (1 Kings 21). None of these episodes can or should be taken at face historical value, but the fact that they are part of the narrative indicates these rulers were weak enough for this to be credible.

And this was common in the era (legend tells of Odysseus also plowing his own field). The big empires had largely collapsed around the year 1200, and were only gradually rebuilt. A king who could rule over a handful of cities was doing quite well for himself.

One final note: kings generally didn't really "rule" in our contemporary sense in this period. They were more like institutionalized bandits than what we would recognize as a proper government. The only people they had real control over were their own military forces: Kings were--above all else--military leaders. The episode where Israel establishes itself as a kingdom turns on the need for a king to organize the local defense. Most of what we would understand as governing was performed by religious or tribal authorities.

Many thanks - that answers a question I’ve had since I was in a religious primary school 30 years ago.