They have the two characteristics of public goods.

Non-excludability. Once you provide flood defences for a city, everyone in that city will benefit and be protected. (People will be protected whether or not they have have contributed towards the cost).
Non-rivalry. If you enjoy a city defended from rising flood waters, it doesn’t reduce the amount of flood defences for other people. (It’s not like eating emergency rations, which do reduce the amount available for other people.)
Failure of the free market

Difficulty in provision. There is little, if any, incentive for a firm to provide flood defences through charging local residents. There is a big free rider problem. People on flood plains may have a vested interest in better flood defences, but there is temptation to avoid paying and hoping someone else will pay.
Another difficulty is that flooding has no predictability. Building roads (a quasi public good) gives a predictable return. Spending money on police and justice also give a predictable return. But, you could spend millions on flood defences and not need it for 30 years.
Another difficulty is that flood defences can be quite difficult to build. Dredging rivers done annually is expensive, though at least manegable. But, other flood defences may be more difficult, such as building relief canals to take excess water. This may need government help to purchase the necessary land.
Occasions where public good can be provided by the free market

Despite the limitations of public goods, it is still possible local residents may club together in a spirit of solidarity to provide local flood defences. Many of the flooding in England is along the affluent banks of the River Thames. Many of these householders are wealthy and their property iss worth a lot (or at least was worth a lot). It is easier to create a private initiative if there are many wealthy people involved, as they will have more spare cash to pay for flood defences, and also they have a bigger vested interest in defending the value of their properties. If the floods affected rented property of low income workers, it would be much harder to provide flood defences. But, these local initiatives may be limited as many flood defence projects require very substantial capital investment.

At a local level, parish councils or local authorities may be able to pay for certain amounts of flood defences.