The photoproduction of eta-mesic 3He has been investigated using the TAPS calorimeter at the Mainz Microtron accelerator facility MAMI. The total inclusive cross section for the reaction gamma3He-->etaX has been measured for photon energies from threshold to 820 MeV. The total and angular differential coherent eta cross sections have been extracted up to energies of 745 MeV. A resonancelike structure just above the eta production threshold with an isotropic angular distribution suggests the existence of a resonant quasibound state. This is supported by studies of a competing decay channel of such a quasibound eta-mesic nucleus into pi(0)pX. A binding energy of (-4.4+/-4.2) MeV and a width of (25.6+/-6.1) MeV is deduced for the quasibound eta-mesic state in 3He.