Inelastic means consumers buy the same amount of some good regardless of the market. That means there's a constant demand.

This is not the case with housing, at least in a local area. Prices go up, people move away and demand goes down. That's elastic.

If housing suddenly cost 10 billion dollars a day in Manhattan, the city would be a ghost town tomorrow. Elastic.

Elasticity of a market is a scale. Elastic markets quickly adjust demand or supply while inelastic markets require a lot of time if they change at all.

Housing on the supply side is relatively inelastic when compared to other markets. Even if demand is high it might not quickly lead to new construction.

On the demand side the market is also relatively inelastic as moving has a lot of friction compared for instance with not making a luxury good purchase or postponing a holiday.