This idea is intended to provide peace of mind for two kinds of people: garden-variety paranoid folks who are scared that *somebody* someday *might* try to access their computer and violate their privacy, and folks who actually have something to hide and are expecting men in black to storm their house
confiscate their files at any moment.
This idea is for a special monitor, self-constructed and self-programmed so it doesn't ever end up on store shelves and tip off the folks you're hiding from that you might be using it.
The monitor (a flatscreen monitor) has a secret compartment under the pedestal. The compartment door is entirely flushmounted and can only be opened by pressing a flushmounted button on the bottom of the screen. Inside the compartment is a keypad, with which you must enter the right passcode.
Once the passcode is entered, another, smaller compartment opens inside the keypad compartment. Inside the smaller compartment is a tiny flashdrive, rigged to perform a complete memory wipe if the door is opened without the passcode. It is on this flashdrive that your most secret files are stored. These files never appear on your computer's harddrive, because they are saved directly to the hidden flashdrive in the monitor, and are not stored in any component of the tower.
This way, when the feds start scouring your computer for the files in question, they'll never find them unless they think to check the monitor itself, which is highly unlikely, figure out what's inside, AND figure out how to safely remove it without the passcode. This provides protection even from that pesky UK law that lets investigators force you to provide all of your encryption keys because it doesn't apply to physical keypads, and in the US, where they cannot force you to do that (we're protected by the 5th Amendment) encryption makes it even MORE unlikely that they'll ever retrieve the files.