Precariousness is a state of being in danger or unsure about something. The precariousness of a small child balancing on a rock wall might cause you to wait below with your arms outstretched, ready to catch her.
A literal kind of precariousness is when your dizzy sister is teetering on a balance beam or your grandmother's favorite flowery teacup looks like it's too close to the edge of the coffee table. There's also the figurative precariousness of being dependent on luck or chance, like the precariousness of graduating from art school with a large amount of debt and no job in sight. The Latin root, precarius, means "obtained by asking or praying."