I have a 500Gb hard drive with 20 bad sectors according to gnome-disk, and when I try to run a SMART self-test, it fails at reading. But gnome-disk shows that every other SMART attribute is OK, and this SMART failure doesn't kick in my motherboard's UEFI SMART failure warning on boot.
Is the hard drive unreliable? Can I still use it safely? Is there anything I can do to fix it or prevent failure?
You should replace your hard drive if you value your data.
SMART in consumer hardware grade is usually not very useful and the firmware mostly reports everything is OK; usually in business/server grade hardware it is more informative.
Bad sectors also usually are masked internally by the hard drive up to a point. By the time they start showing up/being visible to the outside, it is time to dump the media/hard drive and replace it with a new one.
See Google Says Diagnostics Don't Catch Many PC Drive Failures
smartctl /dev/sda -Aas root, so we can actually see the SMART attributes (instead of having to guess) and explain to you what they mean. But if you already have 20 bad sectors, and the SMART self-test encountered a 21st bad sector, it's definitely time to make a backup, and the drive is very likely on its way out.