Atomic isotope shifts (ISs) are the isotope-dependent energy differences between atomic electron energy levels. These shifts have an important role in atomic and nuclear physics, and have been recently suggested as unique probes of physics beyond the standard model under the condition that they are determined significantly more precisely than the current state of the art. In this Letter, we present a simple and robust method for measuring ISs by taking advantage of Hilbert subspaces that are insensitive to common-mode noise yet sensitive to the IS. Using this method we evaluate the IS of the 5S_{1/2}↔4D_{5/2} transition between ^{86}Sr^{+} and ^{88}Sr^{+} with a 1.6×10^{-11} relative uncertainty to be 570 264 063.435(5)(8) (statistical)(systematic) Hz. Furthermore, we detect a relative difference of 3.46(23)×10^{-8} between the orbital g factors of the electrons in the 4D_{5/2} level of the two isotopes. Our method is relatively easy to implement and is indifferent to element or isotope, paving the way for future tabletop searches for new physics, posing interesting prospects for testing quantum many-body calculations, and for the study of nuclear structure.