We show that quantum confinement in the valence and conduction bands of semiconducting single-walled carbon nanotubes can be engineered by means of artificial defects. This ability holds potential for designing future nanotube-based quantum devices such as electrically driven, telecom-wavelength, room-temperature single-photon sources. Intrananotube quantum dots with sub-10 nm lateral sizes are generated between consecutive Ar+ or N+ ion-induced defects, giving rise to quantized electronic bound states with level spacings of the order of 100 meV and larger. Using low-temperature scanning tunneling spectroscopy, we resolve the energy and real space features of the quantized states and compare them with theoretical models. Effects on the states structure due to asymmetric defect scattering strength and the influence of the Au(111) substrate are remarkably well reproduced by solving the Schrödinger equation over a one-dimensional piecewise constant potential model. Using ab-initio calculations, we demonstrate that vacancies, chemisorbed nitrogen ad-atoms and highly stable double vacancies constitute strong scattering centers able to form quantum dots with clear signatures of discrete bound states as observed experimentally. The energy dependence of the defects scattering strength is also studied. Finally, steps toward a characterization of the optical properties of such quantum dot structures are discussed.