L 98-59 is an M3V dwarf star that hosts three recently discovered small planets. The hoststar is bright (K = 7.1) and nearby (10.6 pc), making the system a prime target for follow-upcharacterization with the Hubble Space Telescope (HST) and the upcoming James Webb SpaceTelescope (JWST). The L 98-59 planets orbit close to their star, receiving irradiances that rangefrom 5 to 24 times greater than Earth receives from the Sun. This places the planets in the VenusZone, making our ability to characterize their atmospheres increasingly important inunderstanding the conditions for planetary habitability. Herein, we use simulated transmissionspectroscopy to evaluate the detectability of spectral features with HST/WFC3 andJWST/NIRSpec Prism + MIRI LRS assuming atmospheric compositions that are predominantlycomposed of H2, H2O, CO2, or O2. Due to the relatively deep transits for each planet in thesystem, we find that H2O and CH4 present in a low mean-molecular weight atmosphere could bedetected with HST/WFC3 in one single transit for the two outermost planets, while H2O in aclear steam atmosphere could be detected in 9 transits or fewer with HST/WFC3 for all threeplanets. However, the presence of aerosols would greatly increase the number of transits requiredfor a significant HST detection. We project that observations with JWST would significantlyreduce the number of transits required to detect a clear or cloudy steam atmosphere, while alsobeing able to detect clear, desiccated atmospheres dominated by O2. In the case of a post-runaway atmospheric state, a clear atmosphere dominated by CO2 may be detected in 30 transitsor fewer for each planet, but the presence of haze nearly doubles this requirement. The deeptransits and brightness of the host star make the L 98-59 system an excellent laboratory forcomparative planetary studies of transiting multiplanet systems, and observations of the systemvia both HST and JWST would present a unique opportunity to test the accuracy of the modelspresented in this study.