Sodium can not be stored in water because Reaction of sodium and water is highly exothermic. During this reaction hydrogen gas is produces which catches fire due to heat released during reaction.

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Sodum is a very-very  reactive metal. It reacts vigourously(i.e. it becomes hot and bursts in to flames) when it comes in contact with water.  Since kerosene and water do not mix with each other and kerosene is cheap also, i.e. why sodium is kept under kerosene and not under water. And, yes, if sodium is kept in air(i.e. without any liquid state of water present near it), then also, sodium catches fire as the moisture/water vapour present in the air reacts with the metal and it bursts into flames.

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Elemental sodium does not occur naturally on Earth, because it quickly oxidizes in air and is violently reactive with water, so it must be stored in an inert medium, such as a liquid hydrocarbon. Sodium reacts exothermically with water: small pea-sized pieces will bounce across the surface of the water until they are consumed by it, whereas large pieces will explode so the sodium metal is stored in kerosene(inert liquid hydrocarbon)

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