Found on this page, the author states the origin of the hyperspace sounds (travel while in hyperspace) were created by:

Initially the sound of a McDonnell Douglas DC-9, jet, Burtt tinkered slightly with the recording to bring it to the galaxy far, far away. Taking two copies of the DC-9 sound and playing them slightly out of sync, the product is an oscillating version of the engine.

It doesn't, however, give the origin of going into hyperspace, as you've requested. It appears that this site, however does cover all of the noises of when the Millenium Falcon's hyperdrive conks out:

Eyes on Cinema has dug up an interview from 1980 featuring Ben Burtt – the sound designer for Star Wars, and in the video there is a breakdown of all the various noises that came together to create the clatter that occurs when the Millennium Falcon fails to go into hyperspace in The Empire Strikes Back. Apparently there were eight sounds that wound up being combined to create the effect, but the video only identifies five of them, namely: The inertia starter of an old 1928 biplane

An air jet recorded in a dentist’s office

The sound of an arclight motor starting and stopping

The sound of a motor located in the turret of an armored tank

Pipes underneath a broken sink in the bathroom at the recording studio

Here's the interview which the information comes from:

And here's the clip where it fails with a little better resolution:

It appears that this site has the lock, though. There's an interview with a bunch of the sound people from Star Wars, including Richard Anderson, who I believe was the sound mixer for at least part of the series: