Adding capacitor banks inside conventional PC to compensate for temporary power line disruptions
Capacitors are sometimes used to provide emergency power to devices like SSDs so that the device can store any cached data to non-volatile memory and then shutdown.  See for example:  https://www.intel.com/content/dam/www/public/us/en/documents/technology-briefs/ssd-power-loss-imminent-technology-brief.pdf
Similarly, there are capacitors on your power supply that can provide charge during (very) brief dips in mains voltage.  However, for back up power on a scale of minutes, uninterruptible power supplies use batteries, which store energy more cheaply than capacitors.