Anything that comes closer than the ISS (give or take) is going to interact with the atmosphere which is going to change the dynamics considerably. Generally something with the right orbit to pass 20m above the earth's surface is going to burn up in the atmosphere or hit the surface. The drag is going to kill its orbit fairly quickly.
To have enough energy to come that close without doing so means it has enough energy that superheating the atmosphere or generating nuclear events through impacts with air molecules starts to become a problem (or both: first one then the other). This is the "baseball at the speed of light" type problem from XKCD.
If you come in at exactly the right angle (because you'll skip off the upper atmosphere if you come in too flat) and if you're fast enough, it's entirely possible to enter the atmosphere but fail to complete the aerobreaking maneuver, resulting in an exit from the atmosphere at above escape velocity.
The stress of this maneuver is considerable, especially if you get as low as 20m above ground, so the object would need considerable shear strength and yield strength. Also high density and high thermal capacity. But not unrealistic, I think a tungsten ball (or better yet, a solid tungsten lifting body with aerodynamic steering authority) should make it through.
You can even exit the atmosphere but not have escape velocity, effectively using the aerobreaking maneuver to assist in the gravity capture of your object. But you'd better circularize the orbit shortly after, otherwise your next pass through the atmosphere is going to be terminal.
Relativistic baseball effects aren't very relevant yet, I'm talking about objects hitting the upper atmosphere with around 20-50 km/s. Enough to leave again, not enough to start a fusion reaction.
To have enough energy to come that close without doing so means it has enough energy that superheating the atmosphere or generating nuclear events through impacts with air molecules starts to become a problem (or both: first one then the other). This is the "baseball at the speed of light" type problem from XKCD.