At the moment, we feel kind of connected to many places off-Earth. We have active experiments, orbiters and rovers on the Moon and Mars. Japan's Akatsuki is in orbit around Venus. New Horizons has whizzed past Pluto and into the Kuiper Belt.
But just imagine if we couldn't send anything more into space. This situation might arise if the density of space junk becomes too great and the exponential cascade of collisions causes the Kessler Syndrome. Objects launched from Earth would not escape fatal damage from these hypervelocity collisions.
Gradually, one by one, orbiters would crash onto planetary surfaces as their fuel ran out. Batteries would fail, materials would decay. One by one, the little spacecraft voices that come to the antennas of Earth would fall silent.
The great antennas set to listen to the human sounds of space would cease their turning like sunflowers to catch the signal; they in their turn would become useless monument to the Space Age.