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Extragalactic Fast X-ray Transients (FXRTs) are short flashes of X-ray photons spanning a few seconds to hours, with an uncertain origin. Our ignorance about their physical mechanisms and progenitor systems is due in part to the lack of clear multi-wavelength counterparts in most cases because they only have been identified serendipitously. We develop a systematic search of FXRTs using a straightforward X-ray flare search algorithm, in the Chandra Source Catalog (Data Release 2.0; 169.6 Ms over 592.4 deg$^{2}$ using only observations with $|b|>10^{\circ}$ and before 2015), incorporating various multi-wavelength constraints to rule out Galactic contamination and characterize the candidates. We report the detection of 14 FXRT candidates from a parent sample of 214,701 sources. Candidates have peak 0.5-7 keV fluxes between 1$\times$10$^{-13}$ to 2$\times$10$^{-10}$ erg cm$^{-2}$ s$^{-1}$ and $T_{90}$ values from 4 to 48 ks. The sample can be subdivided into two groups: six "nearby" FXRTs that occurred within $d\lesssim$100 Mpc and eight "distant" FXRTs with likely redshifts $\gtrsim$0.1. Three distant FXRT candidates exhibit light curves with a plateau (${\approx}$1-3 ks duration) followed by a power-law decay and X-ray spectral softening, similar to what was observed for the previously reported FXRT CDF-S~XT2, a proposed magnetar-powered binary neutron star merger event. After applying completeness corrections, we calculate event rates for the nearby and distant samples of 53.7$_{-15.1}^{+22.6}$ and 28.2$_{-6.9}^{+9.8}$ deg$^{-2}$ yr$^{-1}$, respectively. This novel sample of Chandra-detected extragalactic FXRT candidates, although modest in size, breaks new ground in terms of characterizing the diverse properties, nature, and possible progenitors of these enigmatic events.