At each cell division, the spindle self-organizes from microtubules and motors. In human spindles, the motors dynein and Eg5 generate contractile and extensile stress, respectively. Inhibiting dynein or its targeting factor NuMA leads to unfocused, turbulent spindles, and inhibiting Eg5 leads to monopoles; yet, bipolar spindles form when both are inhibited together. What, then, are the roles of these opposing motors? Here, we generate NuMA/dynein- and Eg5-doubly inhibited spindles that not only attain a typical metaphase shape and size but also undergo anaphase. However, these spindles have reduced microtubule dynamics and are mechanically fragile, fracturing under force. Furthermore, they exhibit lagging chromosomes and a dramatic left-handed twist at anaphase. Thus, although these opposing motors are not required for spindle shape, they are essential to its mechanical and functional robustness. This work suggests a design principle whereby opposing active stresses provide robustness to force-generating cellular structures.