Yes, "The astronomers at Palomar say they can see you if they knew where to point the Big Eye."
Nope, that isn't possible. Even using adaptive optics, which removes blurring caused by Earth's atmosphere, the best resolution we can get is about 200 meters per pixel. That's about what we got when we took a good look at the Moon last October as NASA's LCROSS probe crashed into the crater Cabeus:
So if Spaceship Luna (or its shadow), was a few football fields long we would just be able to distinguish something with modern instrumentation. Alas, Spaceship Luna wasn't portrayed as being that big, so technical adviser Robert A. Heinlein didn't quite get it right. (Still, it is a cool film.)
I bring this up because I am often asked if we look at the artifacts left on the Moon by the Apollo astronauts. As with Spaceship Luna, they are too small to be seen from Earth-based telescopes. That isn't true from the vantage point of the Lunar Reconnaissance Orbiter, which has returned images of the Apollo landing sites (for an example see High Noon at Tranquility Base).
On that same vein I am also often asked if we look at the International Space Station. It turns out that the ISS is an easy target and amateur astronomers have taken some amazing images of it (for an example see the shots posted at spaceweather.com).
Yet we don't take any time from our nights to look at the ISS. Why? The Hale Telescope was built not to look at the places and things we already know about. It was built to study the unknown.