Indefinite metacavities (IMCs) made of hyperbolic metamaterials show great advantages in terms of extremely small mode volume due to large wave vectors endowed by the unique hyperbolic dispersion. However, quality ($Q$) factors of IMCs are limited by Ohmic loss of metals and radiative loss of leaked waves. Despite the fact that Ohmic loss of metals is inevitable in IMCs, the radiative loss can be further suppressed by leakage engineering. Here we propose a mirror coupled IMC structure which is able to operate at Fabry–Pérot bound states in the continuum (BICs) while the hyperbolic nature of IMCs is retained. At the BIC point, the radiative loss of magnetic dipolar cavity modes in IMCs is completely absent, resulting in a considerably increased $Q$ factor ($>90$). Deviating from the BIC point, perfect absorption bands ($>0.99$) along with a strong near-field intensity enhancement ($>1.8\times {10}^4$) appear when the condition of critical coupling is almost fulfilled. The proposed BICs are robust to the geometry and material composition of IMCs and anomalous scaling law of resonance is verified during the tuning of optical responses. We also demonstrate that the Purcell effect of the structure can be significantly improved under BIC and quasi-BIC regimes due to the further enhanced $Q$ factor to mode volume ratio. Our results provide a new train of thought to design ultra-small optical nanocavities that may find many applications benefitting from strong light–matter interactions.