In field-effect transistors, the inherent randomness of dopants and other charges is a major cause of device-to-device variability. For a quasi-one-dimensional device such as carbon nanotube transistors, even a single charge can drastically change the performance, making this a critical issue for their adoption as a practical technology. Here we calculate the effect of the random charges at the gate-oxide surface in ballistic carbon nanotube transistors, finding good agreement with the variability statistics in recent experiments. A combination of experimental and simulation results further reveals that these random charges are also a major factor limiting the subthreshold swing for nanotube transistors fabricated on thin gate dielectrics. We then establish that the scaling of the nanotube device uniformity with the gate dielectric, fixed-charge density, and device dimension is qualitatively different from conventional silicon transistors, reflecting the very different device physics of a ballistic transistor with a quasi-one-dimensional channel. The combination of gate-oxide scaling and improved control of fixed-charge density should provide the uniformity needed for large-scale integration of such novel one-dimensional transistors even at extremely scaled device dimensions.