So... is Planetes finished or not?
NO, it is not finished.
Why?
The anime finished after the manga:
Anime finished: April 17, 2004
Manga finished: January 08, 2004
... But MangaUpdates contains information about the anime adaptation and indicates that it is incomplete.
Watching Order (by release date)
Here you have all the anime relations we detected ordered by release date, so you can see the reason for this status in context and also use it as watching order.
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Synopsis
In the year 2075, space travel is commonplace, and corporations have a significant presence in space. Ai Tanabe, a bright-eyed newcomer, joins Technora Corporation's Debris Section, a group responsible for collecting dangerous space junk.
The Debris Section is the least respected and most underfunded department, working with outdated equipment. Ai, however, is determined to make the most of her position and contribute to the future of space exploration.
The story follows Ai's experiences as she navigates her challenging job and gets to know her quirky coworkers, each with their own motivations and dreams related to space. While dealing with the daily grind of debris collection, they also grapple with their personal ambitions and the broader implications of humanity's expansion into space.
Planetes explores the everyday lives of ordinary people working in extraordinary circumstances, focusing on their personal stories against the backdrop of the vast and evolving space frontier.