  Interfacing photonic and solid-state qubits within a hybrid quantum
architecture offers a promising route towards large scale distributed quantum
computing. In that respect, hybrid quantum systems combining circuit QED with
ions doped into solids are an attractive platform. There, the ions serve as
coherent memory elements and reversible conversion elements of microwave to
optical qubits. Among many possible spin-doped solids, erbium ions offer the
unique opportunity of a coherent conversion of microwave photons into the
telecom C-band at $1.54\,\mu$m employed for long distance communication. In our
work, we perform a time-resolved electron spin resonance study of an
Er$^{3+}$:Y$_2$SiO$_5$ spin ensemble at milli-Kelvin temperatures and
demonstrate multimode storage and retrieval of up to 16 coherent microwave
pulses. The memory efficiency is measured to be 10$^{-4}$ at the coherence time
of $T_2=5.6\,\mu$s.
