With hardware resiliency, your internal systems are designed to withstand certain types of mechanical problems without shutting down.
Some common examples would include storing data across multiple RAID disks, so that data could still be recovered if a disk were ever to fail. Hot-swappable drives also allow you to change or upgrade disks without shutting down the system.
In case of a power outage, you may want to consider an Uninterruptable Power Supply (UPS). This will allow your system to continue running for a few minutes until the power comes back on. A UPS will also help prevent lengthy system restarts in the event of quick power flickers.
Another common tactic employed for hardware resiliency is to have redundant network connections in place. This way, you can remain online if one of the connections is severed.