A chemical injection system for use with a subterranean well, the system comprising: a chemical treatment fluid;and a chemical injection valve configured to inject the chemical treatment fluid into an interior of a tubular string in the well, the chemical injection valve comprising: a tubular housing assembly, in which the chemical injection valve receives the chemical treatment fluid via at least one inlet port formed radially through the housing assembly, first and second flow passage sections (38a, 38b), a valve stem reciprocable between closed and open positions to thereby respectively prevent and permit flow of the chemical treatment fluid from the first flow passage section to the second flow passage section, in which the valve stem displaces to the open position in response to an increase in pressure in the first flow passage section to at least a predetermined level, and a lateral bypass passage which communicates with the second flow passage section and an annulus surrounding the valve stem in the closed position of the valve stem, the annulus being positioned longitudinally between a resilient primary seal and a metal secondary seal; in which the valve stem is reciprocably disposed in the housing assembly, and the secondary seal prevents flow through an annular space between the valve stem and the housing assembly. a chemical treatment fluid;and a chemical injection valve configured to inject the chemical treatment fluid into an interior of a tubular string in the well, the chemical injection valve comprising: a tubular housing assembly, in which the chemical injection valve receives the chemical treatment fluid via at least one inlet port formed radially through the housing assembly, first and second flow passage sections (38a, 38b), a valve stem reciprocable between closed and open positions to thereby respectively prevent and permit flow of the chemical treatment fluid from the first flow passage section to the second flow passage section, in which the valve stem displaces to the open position in response to an increase in pressure in the first flow passage section to at least a predetermined level, and a lateral bypass passage which communicates with the second flow passage section and an annulus surrounding the valve stem in the closed position of the valve stem, the annulus being positioned longitudinally between a resilient primary seal and a metal secondary seal; in which the valve stem is reciprocably disposed in the housing assembly, and the secondary seal prevents flow through an annular space between the valve stem and the housing assembly. a tubular housing assembly, in which the chemical injection valve receives the chemical treatment fluid via at least one inlet port formed radially through the housing assembly, first and second flow passage sections (38a, 38b), a valve stem reciprocable between closed and open positions to thereby respectively prevent and permit flow of the chemical treatment fluid from the first flow passage section to the second flow passage section, in which the valve stem displaces to the open position in response to an increase in pressure in the first flow passage section to at least a predetermined level, and a lateral bypass passage which communicates with the second flow passage section and an annulus surrounding the valve stem in the closed position of the valve stem, the annulus being positioned longitudinally between a resilient primary seal and a metal secondary seal; in which the valve stem is reciprocably disposed in the housing assembly, and the secondary seal prevents flow through an annular space between the valve stem and the housing assembly.