We have obtained spectroscopy of Q1343 + 266A, B, a pair of quasars at redshift z = 2.03 with a projected separation of 9''.5. This system is well suited for probing the Ly alpha forest, since the two-component spectra show several Ly alpha lines in common and several others not. Using Bayesian statistics, under the idealization of uniform-radius spherical absorbers, we find that the Ly alpha cloud radius at z approximate to 1.8 lies in the range 40 h(-1) kpc < R < 280 h(-1) kpc with 98% confidence in a (Omega(0), Lambda(0)) = (1, 0) model universe, where h drop (H-0/100 km s(-1) Mpc(-1)). The median value of R is 90 h(-1) kpc. These numbers scale up by a factor 1.44 for (Omega(0), Lambda(0)) = (0.1, 0) and by a factor 1.85 for (0.1, 0.9). Pressure-confined and freely expanding Ly alpha cloud models in which the absorbers formed in a significantly more compressed state at z >> 2 are contradicted by these new data, as are models involving stably confined gas concentrations in photoionization equilibrium within minihalos of cold dark matter. The comoving density of Ly alpha forest objects at z similar to 2 is similar to 0.3 h (R/100 kpc)(-2) Mpc(-3) in a (1, 0) model universe. This suggests a possible identification of Ly alpha clouds with the unvirialized, collapsing progenitors of the faint, blue galaxies at z similar to 1.