We experimentally demonstrate a microwave quantum storage for two spectral modes of microwave radiation in an on-chip system of eight coplanar superconducting resonators. Single-mode storage shows a power efficiency of up to 60±3% at single photon energy and more than 73±3% at higher intensity. The noiseless character of the storage is confirmed by coherent-state quantum process tomography. The demonstrated efficiency is an order of magnitude higher than the previously reported data for multimode microwave quantum memory. The proposed on-chip quantum memory architecture can be easily integrated into the state-of-the-art superconducting quantum circuit technology without any coherence compromises. The obtained results are in good agreement with the proposed theory, which pave the way for building a practical multimode microwave memory for superconducting quantum circuits.