Can all areas on Earth experience a total solar eclipse? A solar eclipse only puts part of Earth at a time in shadow, so I'm not talking about a solar eclipse that casts a shadow over all of Earth.

What I mean is: are there any areas on Earth where, no matter how Sol, Earth and Luna are aligned, a total solar eclipse is impossible? Can the penumbra cast a shade on any part of Earth, or are there certain areas where a penumbra due to a solar eclipse cannot occur?
Yes, total Solar eclipses can occur anywhere. On any given place, they occur roughly every 400 years, on average. If you take a look on the picture below (from Wikipedia), you'll see that within the last one thousand year, almost every place on Earth has experienced a total Solar eclipse. The places that haven't experienced one have no special latitude or longitude, so it's just a matter of coincidence.