YBa2Cu3O7 grain boundary junctions and low-noise superconducting quantum interference devices patterned by a focused ion beam down to 80 nm linewidth

YBa$_2$Cu$_3$O$_7$ 24$^\circ$ (30$^\circ$) bicrystal grain boundary junctions (GBJs), shunted with 60\,nm (20\,nm) thick Au, were fabricated by focused ion beam milling with widths $80\,{\rm nm} \le w \le 7.8\,\mu$m. At 4.2\,K we find critical current densities $j_c$ in the $10^5\,{\rm A/cm^2}$ range %\dkc{\#1} (without a clear dependence on $w$) and an increase in resistance times junction area $\rho$ with an approximate scaling $\rho\propto w^{1/2}$. For the narrowest GBJs $j_c\rho\approx 100\,\mu$V, which is promising for the realization of sensitive nanoSQUIDs for the detection of small spin systems. We demonstrate that our fabrication process allows the realization of sensitive nanoscale dc SQUIDs; for a SQUID with $w\approx 100$\,nm wide GBJs we find an rms magnetic flux noise spectral density of $S_\Phi^{1/2}\approx 4\,\mu\Phi_0/{\rm Hz}^{1/2}$ in the white noise limit. We also derive an expression for the spin sensitivity $S_\mu^{1/2}$, which depends on $S_\Phi^{1/2}$, on the location and orientation of the magnetic moment of a magnetic particle to be detected by the SQUID, and on the SQUID geometry. For the not optimized SQUIDs presented here, we estimate $S_\mu^{1/2}=390\,\mu_B/\sqrt{\rm{Hz}}$, which could be further improved by at least an order of magnitude.