All I know is that it's basically about a little girl who sneaks onto a spaceship. The ship can't land because its too heavy, all automated or something. But she is found and they try to shoot a bunch of junk out into space to get the weight right, no joy, so at the end she calls up her parents that are on another ship, says goodbye, and I guess she gets shot out into space.
That's about the gist of it. Not sure if it's from The Outer Limits or The Twilight Zone.