Yes and no--how's that for an ambiguous answer? Really, though, here's why the story is credible.  As science and technology allow humans to move past long-established boundaries, we often don't have time to consider the moral implications that breeching those boundaries might produce.  In a future world, where the impossible is possible, teens such as Madeline will be used to being able to go wherever they want to go and space travel might seem like an everyday occurrence.  When Madeline boards the ETS, she does not consider that because humans are not made for space travel, certain rules and regulations must be enforced in order to assure safety.

Likewise, the story's theme of a person having to make a difficult life-or-death decisions is a universal one and one that seems even more plausible in a futuristic setting.

In spite of those realistic elements, readers must question how a society that is so advanced that it has humans living on many different planets cannot design an efficient spacecraft that could carry a fuel reserve and another passenger.  Also, it seems unlikely that Madeline just happens to stowaway on the one spacecraft that is on a desperate mission to deliver medicine and cannot be diverted.

In "The Cold Equations," is Godwin's space frontier believable?

The futuristic nature of this excellent short story means that Godwin is speculating about possible futures, and it is clear that the setting presents us with just one of many possible scenarios of the future world that he imagines. His setting is incredibly bleak and unyielding, as we are presented with a world in which the difficulties of maintaining contact between outlying space colonies results in the "cold equations" of the story's title becoming incredibly important:

The cruisers were forced by necessity to carry a limited amount of bulky rocket fuel, and the fuel was rationed with care, the cruiser's computers determining the exact amount of fuel each EDS would require for its mission. The computers considered the course coordinates, the mass of the EDS, the mass of pilot and cargo; they were very precise an daccurate an domitted nothing from their calculations.

On the one hand, such a scenario does seem a bit to scientifically bleak to me. Surely we think about progress and the future with a more optimistic air than the rather depressing world Godwin presents us with. However, on the other hand, we need to remember that in a future world where space travel is the norm, the logistics of maintaining contact between distant colonies would represent a massive challenge, and thus this story could be argued to be an accurate reflection of some of those challenges.