Lifetime spectra of positrons have been measured for acrylic epoxy-based network polymers. For the specimens with the different permeability coefficients to water vapor C-p, the lifetime of ortho-positronium (o-Ps) tau(3) increased with increasing C-p. This fact suggests that the permeability increases with an increase in the size of open spaces. From measurements of temperature dependencies of tau(3) and the intensity of o-Ps, three onset temperatures for the change in the temperature gradient of these parameters were determined. The highest onset temperature (T-alpha = 300-325 K) was identified to be the glass transition temperature, and others (T-gamma = 90-180 K and T-beta = 160-205 K) were associated with the onset temperatures for limited local motions of molecules; those molecular mot;ions were found to be affected by both the number of crosslinks and the presence of free side chains.