We realize simultaneous quantum degeneracy in mixtures consisting of the alkali and alkaline-earth-like atoms Li and Yb. This is accomplished within an optical trap by sympathetic cooling of the fermionic isotope ⁶Li with evaporatively cooled bosonic ¹⁷⁴Yb and, separately, fermionic ¹⁷³Yb. Using cross-thermalization studies, we also measure the elastic s-wave scattering lengths of both Li-Yb combinations, |a(⁶Li-¹⁷⁴Yb)| = 1.0 ± 0.2 nm and |a(⁶Li-¹⁷³Yb)| = 0.9 ± 0.2 nm. The equality of these lengths is found to be consistent with mass-scaling analysis. The quantum degenerate mixtures of Li and Yb, as realized here, can be the basis for creation of ultracold molecules with electron spin degrees of freedom, studies of novel Efimov trimers, and impurity probes of superfluid systems.