Star Eagle
The Star Eagle is a ship gifted to players who complete the eight-mission Freestar Collective questline. To begin the faction questline, head to Akila City and look for the people dressed up like wild west sheriffs. Upon completing the final quest, called The Hammer Falls, you'll have earned the right to pilot this gem.
UC Prison Shuttle
The UC Prison Shuttle is a starship you'll come into possession of as part of the Crimson Fleet questline, which will begin the first time you get caught committing a crime in UC controlled space (like New Atlantis). Follow this thread all the way to the "Echoes of the Past" storyline and you'll get this silly ship as a prize.
Wanderwell
To get this ship, you'll need to select the Kid Stuff trait during character creation. Once you begin your playthrough, visit your parents at their apartment in New Atlantis and continue checking in with them every once in awhile to advance their questline. At the conclusion of the story, you'll receive Wanderwell as a gift -- well worth the cost of sending money away to mom and pop every week.
Starborn Guardian
Without going into spoilers, this ship is obtained by completing Starfield's campaign and beginning anew with New Game+. To say more would go into details you may not wish to know, so we'll leave it at that.
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