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 $rhopropto w^{1/2}$. For the narrowest GBJs $j_crhoapprox 100,mu$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 $wapprox 100$,nm wide GBJs we find an rms magnetic flux noise spectral density of $S_Phi^{1/2}approx 4,muPhi_0/{rm Hz}^{1/2}$ 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.
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