  We search for an isotropic stochastic gravitational-wave background (GWB) in
the newly released $11$-year dataset from the North American Nanohertz
Observatory for Gravitational Waves (NANOGrav). While we find no significant
evidence for a GWB, we place constraints on a GWB from a population of
supermassive black-hole binaries, cosmic strings, and a primordial GWB. For the
first time, we find that the GWB upper limits and detection statistics are
sensitive to the Solar System ephemeris (SSE) model used, and that SSE errors
can mimic a GWB signal. We developed an approach that bridges systematic SSE
differences, producing the first PTA constraints that are robust against SSE
uncertainties. We thus place a $95\%$ upper limit on the GW strain amplitude of
$A_\mathrm{GWB}<1.45\times 10^{-15}$ at a frequency of $f=1$ yr$^{-1}$ for a
fiducial $f^{-2/3}$ power-law spectrum, and with inter-pulsar correlations
modeled. This is a factor of $\sim 2$ improvement over the NANOGrav $9$-year
limit, calculated using the same procedure. Previous PTA upper limits on the
GWB will need revision in light of SSE systematic uncertainties. We use our
constraints to characterize the combined influence on the GWB of the stellar
mass-density in galactic cores, the eccentricity of SMBH binaries, and
SMBH--galactic-bulge scaling relationships. We constrain cosmic-string tension
using recent simulations, yielding an SSE-marginalized $95\%$ upper limit on
the cosmic string tension of $G\mu < 5.3\times 10^{-11}$---a factor of $\sim 2$
better than the published NANOGrav $9$-year constraints. Our SSE-marginalized
$95\%$ upper limit on the energy density of a primordial GWB (for a
radiation-dominated post-inflation Universe) is
$\Omega_\mathrm{GWB}(f)h^2<3.4\times10^{-10}$.
