Which classical structure did the Romans discover would self-heal when cracked by water?

Concrete. Roman marine concrete has stood in seawater for two thousand years while modern reinforced concrete has a design life of around a century, and the reason turns out to be a manufacturing habit that looked for a long time like carelessness.

Roman concrete contains small white lumps of lime, and the assumption was poor mixing. A 2023 study argued they are deliberate, produced by hot mixing with quicklime, and that they are a repair mechanism: when water gets into a crack it dissolves the nearby lime clast and recrystallises calcium carbonate across the gap, sealing it. Samples deliberately cracked in the laboratory sealed themselves within two weeks; controls without the clasts did not.

Seawater makes it stronger rather than weaker, growing rare interlocking crystals in the volcanic ash over decades. Modern concrete fails for a different reason entirely - the steel reinforcement inside it rusts, expands and splits the concrete from within - so it is not a fair comparison, and unreinforced Roman concrete could never span what a modern beam spans. Concrete production is also responsible for something like 8 per cent of global carbon emissions, which is why durability research has stopped being an antiquarian interest.