What?
http://scifi.stackexchange.com/questions/50085/how-big-is-the-known-star-trek-universe
If by single solar system you mean huge part of a galaxy then yes - it took part in a single solar system.
If you break up star trek by each individual version it does not represent an entire galaxy even if is the back drop to the story.
According to Captain Picard, in 2373 the Federation spans 12,000 ly. most episodes were spent in one system at a time mostly repeating systems within fed space. DS9 is a different story.
A review of this information identifies some problems. The listed stars include sites of exploration by the crew of the Enterprise and Enterprise-D, implying that they are on the frontiers of Federation space. However, all but two of the stars are within 420 ly of Earth, and half are within 80 ly of Earth. This is a tiny fraction of the volume of Federation space implied by Captain Picard's statement in ST: Generations that the Federation spans 12,000 ly.
http://www.johnstonsarchive.net/startrek/stgeogr.html
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