  We analyzed ROSAT PSPC images of the bright, nearby $(z<0.1)$ galaxy clusters
A401, A1656 (Coma), A2029, A2199, and A2256 to constrain their intrinsic shapes
and baryon fractions. Following Buote & Tsai we probed the aggregate structure
of the clusters on scales $\sim 1.5h^{-1}_{80}$ Mpc to reduce effects of
possible substructure on smaller scales ($\lesssim$ a few hundred kpc). The
ellipticities of the X-ray isophotes are typically $\epsilon\approx 0.15 -
0.25$ and display negative radial gradients significant (except Coma). By
assuming hydrostatic equilibrium and a variety of mass models we obtain
ellipticities $\epsilon_{mass} \approx 0.40-0.55$ for isothermal models of the
total gravitating matter of the clusters; the $\epsilon_{mass}$ constraints
change by $<10%$ upon consideration of the small temperature gradients shown by
ASCA to be typical for rich clusters; the observed X-ray ellipticity gradients
require that mass models with constant ellipticity have steep density profiles,
$\rho\propto r^{-4}$. Estimates of the gas masses are highly insensitive to the
ellipticities of the X-ray isophotes. The clusters in our sample have
increasing fractions of gas mass to total mass with radius and have
$M_{gas}/M_{tot} = (4% - 11%)h^{-3/2}_{80}$ within a radius $1.5h^{-1}_{80}$
Mpc, in excellent agreement with the results of White & Fabian and the Baryon
Catastrophe proposed by White et al. (1993). Finally, the ellipticities of the
dark matter distributions are essentially identical to $\epsilon_{mass}$ and
are consistent with the shapes of dark halos predicted by N-body simulations
and the shapes of the galaxy isopleths in the clusters in contrast to our
previous conclusions using {\it Einstein} data.
