It is generally taken for granted that our Universe is free of antimatter objects and domains. This certitude has recently been challenged by the possible detection of anti-helium nuclei by AMS-02. Should the observation be confirmed, the existence of nearby antistars would make a plausible hypothesis to explain the origin of the antinuclei. In this paper we use the 10-years Fermi Large Area Telescope (LAT) gamma-ray source catalog to set constraints on the abundance of antistars around the Sun. We identify in the catalog 14 antistar candidates not associated with any objects belonging to established gamma-ray source classes and with a spectrum compatible with baryon-antibaryon annihilation. We use them along with an estimate of the LAT sensitivity to antistars to set upper limits on the local antistar fraction $f_{\bar{\ast}}$ with respect to normal stars. We provide parametric limits as a function of the closest antistar mass, velocity, and surrounding matter density. We also employ a novel Monte~Carlo method to set limits for a few hypotheses about the antistar population. For a population with properties equivalent to those of regular stars concentrated in the Galactic disk we obtain $f_{\bar{\ast}} < 2.5 \times 10^{-6}$ at 95\% confidence level, which is 20 times more constraining than limits previously available. For a primordial population of antistars distributed in the Galactic halo we obtain new local upper limits which decrease as a function of antistar mass $M$ from $f_{\bar{\ast}} < 0.2$ at 95\% confidence level for $M = 1 \; M_\odot$ to $f_{\bar{\ast}} < 1.6 \times 10^{-4}$ at 95\% confidence level for $M = 10 \; M_\odot$. By combining these limits with existing microlensing constraints for lighter objects in the Magellanic clouds, we infer that a primordial population of halo antistars must have a density lower than $\mathcal{O}(10^{-5}\;\text{pc}^{-3})$ to $\mathcal{O}(10^{-2}\;\text{pc}^{-3})$ depending on their masses. Our limits can constrain models for the origin and propagation of antinuclei in cosmic rays.