  The introduction of an optical resonator can enable efficient and precise
interaction between a photon and a solid-state emitter. It facilitates the
study of strong light-matter interaction, polaritonic physics and presents a
powerful interface for quantum communication and computing. A pivotal aspect in
the progress of light-matter interaction with solid-state systems is the
challenge of combining the requirements of cryogenic temperature and high
mechanical stability against vibrations while maintaining sufficient degrees of
freedom for in-situ tunability. Here, we present a fiber-based open
Fabry-P\'{e}rot cavity in a closed-cycle cryostat exhibiting ultra-high
mechanical stability while providing wide-range tunability in all three spatial
directions. We characterize the setup and demonstrate the operation with the
root-mean-square stability of less than $90$ pm at temperature of $6.5$ K and
integration bandwidth of $100$ kHz. Finally, we benchmark the cavity
performance by demonstrating the strong-coupling formation of
exciton-polaritons in monolayer WSe$_2$ with a cooperativity of $1.6$. This set
of results manifests the open-cavity in a closed-cycle cryostat as a versatile
and powerful platform for low-temperature cavity QED experiments.
