Under which conditions do the dynamics and thermodynamics of open chemical reaction networks (CRNs), grounded on the notion of idealized chemostats that exchange selected species, emerge from underlying closed CRNs? While open CRNs provide the standard framework to describe out-of-equilibrium chemical systems, real systems are finite and ultimately relax to equilibrium, leaving the status of this description conceptually unresolved. Here, we address this fundamental question. We show that open-CRN behavior arises as an asymptotic regime of closed CRNs when two physically minimal and transparent conditions are met: a time-scale separation, whereby fast reactions effectively act as exchange mechanisms, and an abundance separation, whereby a subset of species behaves as chemostats with diverging chemical capacity. In this regime, both the stochastic dynamics and the thermodynamic structure - including local detailed balance, entropy production, and free-energy balance - emerge to leading order from the underlying closed CRN. We further predict that this regime persists up to a characteristic time proportional to the scale of the abundance separation, thereby providing an experimentally accessible criterion for estimating how long a finite closed system behaves as an open CRN. Our results apply to arbitrary stoichiometries. They show that chemostats need not be introduced as external idealizations, but instead arise as emergent thermodynamic structures within closed systems, providing a unified and physically grounded foundation for the nonequilibrium thermodynamics of CRNs.