Non-space was a void between dimensions.
Non-space was an abyss of infinite nothingness, where time did not exist. Lister compared it to Rimmer's Hammond organ recital nights. It could be accessed via rents in the spacetime continuum caused by a temporal rift. This happened to Starbug when it encountered such a rift in space.
The rift caused a hyperway through non-space to materialise on Starbug's engineering deck. The hyperway was an unstable tunnel which led to a parallel universe. Parts of the link were fragile enough that they could be pierced by The Cat's foot. The hole bled into non-space, something Kryten warned them to avoid.
The Starbug crew met their alternate selves coming in the opposite direction. When it was established that it was Kochanski who survived the accident in their reality, Lister was despondent. At that moment a Kinitawowi battle cruiser managed to infiltrate non-space and attack the hyperway. The link was separated with Kochanski on the wrong side.
She worked frantically to re-establish the hyperway by using electromagnetic phasing frequencies. Kryten assisted her and they were able to reconnect with the alternate Starbug. After celebrating and swapping supplies, Kochanski went to rejoin her shipmates. The GELFs returned and severed the link once more, so Kochanski decided to jump the gap...and missed.
As she fell into infinite non-space, she used a communicator to contact Lister. He quickly found a crossbow and some rope and managed to shoot a line down to her. Unfortunately the crossbow bolt stabbed into Kochanski's leg and she then subjected Lister to several minutes of gratuitous verbal abuse. The hyperlink self-repaired shortly thereafter, leaving Kochanski stuck on the wrong Starbug. (RD: Ouroboros)
Kochanski's plan to open another link to her dimension failed when Lister insisted on passing through a comet's tail to get to another anomaly. Starbug was so badly damaged that they missed the opportunity to make another connection. (RD: Blue)