Is it possible to enable RAID 1 on an already in-use HP Proliant DL380 G5

Yes, this is possible.

See: Can I convert a 1 disk RAID 0 to RAID 1?

You're lucky that your striped array has worked until now. The process you'll go through is an "array transformation".

• Make sure your system has a battery-backed cache (BBWC) on the RAID controller.
• Add two more disks of equal or greater size than the existing disks.
• Install the HP Smart Storage Administrator (HPSSA) software for Windows (or the HP Array Configuration Utility is this is an old version of Windows).
• In the software, add the two new drives to the existing array.
• In the software, perform an array transformation from RAID0 to RAID 1+0.