Previous studies have found that the elemental abundances of a star correlate directly with its age and metallicity. Using this knowledge, we derive ages for a sample of 250,000 stars taken from GALAH DR3 using only their overall metallicity and chemical abundances. Stellar ages are estimated via the machine learning algorithm $XGBoost$, using main sequence turnoff stars with precise ages as our input training set. We find that the stellar ages for the bulk of the GALAH DR3 sample are accurate to 1-2 Gyr using this method. With these ages, we replicate many recent results on the age-kinematic trends of the nearby disk, including the age-velocity dispersion relationship of the solar neighborhood and the larger global velocity dispersion relations of the disk found using $Gaia$ and GALAH. The fact that chemical abundances alone can be used to determine a reliable age for a star have profound implications for the future study of the Galaxy as well as upcoming spectroscopic surveys. These results show that the chemical abundance variation at a given birth radius is quite small, and imply that strong chemical tagging of stars directly to birth clusters may prove difficult with our current elemental abundance precision. Our results highlight the need of spectroscopic surveys to deliver precision abundances for as many nucleosynthetic production sites as possible in order to estimate reliable ages for stars directly from their chemical abundances. Applying the methods outlined in this paper opens a new door into studies of the kinematic structure and evolution of the disk, as ages may potentially be estimated for a large fraction of stars in existing spectroscopic surveys. This would yield a sample of millions of stars with reliable age determinations, and allow precise constraints to be put on various kinematic processes in the disk, such as the efficiency and timescales of radial migration.