We examine the stability of feedback-regulated star formation (SF) in galactic nuclei and contrast it to SF in extended discs. In galactic nuclei the dynamical time becomes shorter than the time over which feedback from young stars evolves. We argue analytically that the balance between stellar feedback and gravity is unstable in this regime. We study this using numerical simulations with pc-scale resolution and explicit stellar feedback taken from stellar evolution models. The nuclear gas mass, young stellar mass, and SFR within the central ~100 pc (the short-timescale regime) never reach steady-state, but instead go through dramatic, oscillatory cycles. Stars form until a critical surface density of young stars is present (such that feedback overwhelms gravity), at which point they begin to expel gas from the nucleus. Since the dynamical times are shorter than the stellar evolution times, the stars do not die as the gas is expelled, but continue to push, triggering a runaway quenching of star formation in the nucleus. However the expelled gas is largely not unbound from the galaxy, but goes into a galactic fountain which re-fills the nuclear region after the massive stars from the previous burst cycle have died off (~50 Myr timescale). On large scales (>1 kpc), the galaxy-scale gas content and SFR is more stable. We examine the consequences of this episodic nuclear star formation for the Kennicutt-Schmidt (KS) relation: while a tight KS relation exists on ~1 kpc scales in good agreement with observations, the scatter increases dramatically in smaller apertures centered on galactic nuclei.