Localized turbulence bounded by non-turbulent flow in a uniformly stratified environment is studied with direct numerical simulations of stably stratified shear layers. Of particular interest is the turbulent/non-turbulent interfacial (TNTI) layer, which is detected by identifying the turbulent region in terms of its potential vorticity. Fluid near the outer edge of the turbulent region gains potential vorticity and becomes turbulent by diffusion arising from both viscous and molecular effects. The flow properties near the TNTI layer change depending on the buoyancy Reynolds number near the interface,$Re_{bI}$. The TNTI layer thickness is approximately 13 times the Kolmogorov length scale for large$Re_{bI}$($Re_{bI}\gtrsim 30$), consistent with non-stratified flows, whereas it is almost equal to the vertical length scale of the stratified flow,$l_{vI}=l_{hI}Re^{-1/2}$(here$l_{hI}$is the horizontal length scale near the TNTI layer, and$Re$is the Reynolds number), in the low-$Re_{bI}$regime ($Re_{bI}\lesssim 2$). Turbulent fluid is vertically transported towards the TNTI layer when$Re_{bI}$is large, sustaining the thin TNTI layer with large buoyancy frequency and mean shear. This sharpening effect is weakened as$Re_{bI}$decreases and eventually becomes negligible for very low$Re_{bI}$. Overturning motions occur near the TNTI layer for large$Re_{bI}$. The dependence on buoyancy Reynolds number is related to the value of$Re_{bI}$near the TNTI layer, which is smaller than the value deep inside the turbulent core region. An imprint of the internal gravity waves propagating in the non-turbulent region is found for vorticity within the TNTI layer, inferring an interaction between turbulence and internal gravity waves. The wave energy flux causes a net loss of the kinetic energy in the turbulent core region bounded to the TNTI layer, and the amount of kinetic energy extracted from the turbulent region by internal gravity waves is comparable to the amount dissipated in the turbulent region.