We present results on the stretching of single tubular vesicles in an elongation flow toward dumbbell shapes, and on their relaxation. A critical strain rate epsilonc exists; for strain rates epsilon<epsilonc, the vesicle remains tubular but fluctuates, though its steady state extension increases with the strain rate epsilon. Above epsilonc, first a shape transition to dumbbell occurs, and then high order shape modes become unstable, leading to a pearling state. We have quantitatively characterized the transition and found a scaling of epsilonc with the system parameters. A remarkable feature of vesicle tube behavior around the critical point is a slowdown of the vesicle relaxation to the final extended state in the vesicle stretching. Such feature is similar to that found in continuous phase transitions and to the critical effects recently observed for polymer molecules near the coil-stretch transition in elongation flow.