Is it possible to hear sounds in space?
The short answer is no.
Sound waves require a medium, like air or water, to travel though. When we speak, for instance, the vibration of our vocal chords compress the air around them. The compressed air then moves the air around it and so on.
Eventually these compressions reach the ears of a listener, and their brain interprets the compression as sound. But without anything to compress, these sound waves can not be transmitted.
In the vacuum of space there is no medium to transmit these sound waves, therefore in the emptiness of space, you would not be able to hear anything.
What About Light?
Light waves are different. Known as electromagnetic radiation, light waves do not require the existence of a medium in order to propagate. (Though the presence of a medium does affect the waves, in particular their path changes when they intersect the medium and they also slow down.)
So light can travel through the vacuum of space unimpeded. This is why we can see distant objects like planets, stars and galaxies.
Haven't Probes Picked Up Sounds From the Planets?
This is a bit of a tricky one. NASA, back in the early 90s, released a five volume set of space sounds. Unfortunately they were none too specific about how the sounds were made exactly.
Only the recordings weren't actually of sound coming from those planets. What was picked up where interactions of charged particles in the planets' magnetospheres, trapped radio waves and other electromagnetic disturbances.
NASA then took these measurements and converted them into sounds. It is similar to the way that your radio captures the radio waves (which are long wavelength light waves) from radio stations and converts those signals into sound.
Didn't Some of the Apollo Astronauts Report Hearing Sounds On and Around the Moon?
There hasn't ever really been an "official" explanation of these sounds (described as whistling and buzz-saw sounds). But most scientists offer that it could be attributed to either radio interference in the lander or perhaps an artifact of the Sun's solar wind.
Since the most prominent example of this sound was when the Apollo 15 astronauts were on the far side of the Moon, it could be suggested that the Moon's gravity was gravitationally focusing the Sun's wind (a mix of high energy charged particles) onto the capsule. That interaction would created electromagnetic distortions, which could produce sounds inside the capsule.
As I said, there is no "official" explanation to this one, but in the context of the sound in space, even here the sounds were only heard inside the lander where there was air to propagate the sound.