Actin filaments inside endothelial cells assemble into tensed bundles called stress fibers. Stress fibers are observed in vivo in a variety of tissues including the aorta, the heart, the spleen, the eye, and hair follicles. Sarcomeres are the force generating units of stress fibers and are responsible for generating intracellular tension. Little is known about the mechanical behavior of individual sarcomeres in living cells. Using femtosecond laser ablation to sever individual stress fibers in living capillary endothelial cells, we are able to measure the mechanics of sarcomeres in living cells. Our results indicate that the length of a sarcomere after severing decreases in two phases- an initial elastic response, followed by slower contraction at constant speed. The latter phase, interpreted as active myosin-mediated contraction, ceases abruptly after a minimum sarcomere length is achieved, suggesting a rigid resistance that prevents further contraction. We model this with an equivalent mechanical circuit, allowing us to estimate the speed of myosin motor walking in sarcomeres for the first time. We find that this speed ranges from 0.02 to 0.1 microns/s, which compares well with in vitro measurements. Our analysis suggests a novel mechanical model of a sarcomere that includes an active force generating component in parallel with an infinite barrier and in series with a stiff elastic spring.