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 $A_mathrm{GWB}<1.45times 10^{-15}$ 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 $Gmu < 5.3times 10^{-11}$---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 $Omega_mathrm{GWB}(f)h^2<3.4times10^{-10}$.
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