You could use mysql or open source equivalent of mariadb.
Then setup replication..caveat both computer have to be on so replication can happen.
If you want a true turn computer B on when computer A fails then the data has to live on another computer. You can setup a network share or just use native networking like mariadb or etc. You can literally load mariadb most operating system, and the computer it is on can be any computer. Especially with such a tiny database, a user using a PC with mysql on it would never even notice a database running 5 tables running in the background. Unless you start having hundreds of thousands of records.
Somewhere the data lives on a hard drive and that is your key point of failure.
Every other part of the computer doesn't matter! CPU dies replace it. done. RAM dies, replace it done. You can even install a database like mariadb on virtually an windows,mac,linux PC and copy the data over from the dead machines hard drive by connecting it to a new working PC via USB external enclosure. However, if the hard drive fails...game over.
You can reduce the odds in fails by swapping the hard drive for an SSD. Hard drive have moving parts subject to failure and SSD don't. SSD will still eventually fail, but you have eliminate one major factor in hard drive reliability.
Unfortunately the rest of standard practices cost significantly more.
RAID 5/6 can protect against single/dual hard drive failure but require 3/4 hard drives minimum.
You will still need backup with versioning to protect against a virus encrypting your files.
You can even up your game further by making it web based. Then in can be accessed from any PC. You can use the built-in windows firewall to block any IP you don't want accessing the PC. You can load apache,mariadb, and php on windows and then there is nothing to install, except on the server(can be any standard PC). Using a PHP framework like laravel(or any) you can make a beautiful GUI for the client to use. Many frameworks even make it possible to implement username/password logins easily for protection.