  Er$^{3+}$$:$Y$_2$SiO$_5$ is a material of particular interest due to its
suitability for telecom-band quantum memories and quantum transducers
interfacing optical communication with quantum computers working in the
microwave regime. Extending the coherence lifetimes of the electron spins and
the nuclear spins is essential for implementing efficient quantum information
processing based on such hybrid electron-nuclear spin systems. The electron
spin coherence time of Er$^{3+}$$:$Y$_2$SiO$_5$ is so far limited to several
microseconds, and there are significant challenges in optimizing coherence
lifetimes simultaneously for both the electron and nuclear spins. Here we
perform a pulsed-electron-nuclear-double-resonance investigation for an
Er$^{3+}$-doped material at subkelvin temperatures. At the lowest working
temperature, the electron spin coherence time reaches 290 $\pm$ 17 $\mu$s,
which has been enhanced by 40 times compared with the previous results. In the
subkelvin regime, a rapid increase in the nuclear spin coherence time is
observed, and the longest coherence time of 738 $\pm$ 6 $\mu$s is obtained.
These extended coherence lifetimes could be valuable resources for further
applications of Er$^{3+}$$:$Y$_2$SiO$_5$ in fiber-based quantum networks.
