Since you cannot get the newest updates from the Play Store, you must manually download the compatible "legacy" version. Use trusted repositories to avoid malware: APKMirror - Google Play Services 21.02.14
When you need a fuller solution: GApps and custom ROMs
: Before installing, go to Settings > Apps > Google Play Services and select Clear Cache and Clear Data to prevent installation conflicts. How To Fix Google Play Store Not Working
Search for version 21.30.99 . This is widely considered the safest third-party source because they verify cryptographic signatures to ensure files haven't been tampered with.
Finding a working "Google Play Services APK for Android 4.2.2" is a victory against planned obsolescence. It allows a device that should have been recycled a decade ago to still check email, still navigate via GPS, and still function.
Google Play Services is the "glue" that holds your Android experience together. It handles: : Keeps your existing apps from crashing.