It was never known for certain, later, what caused the spaceship to explode on takeoff (or technically, as it attempted to take off).  The fuel which had been prepared for it, had not been as thoroughly tested as normally would be required; perhaps some contaminant was present that caused it to combust in a different way than was expected by the ship's designers.  The o-rings were, in some cases, organic; the leading theory among Martians was that they had been eroded by plastophages, causing a leak of some sort that led to disaster.  This would only make sense if the plastophage contamination had been present from the time it left Earth (so that it had enough time to corrode the o-rings until they failed), which might explain why the surviving Earthlings were more skeptical of this theory (as they were more certain that the plastophages were from Mars, and could not have come from the ship).

What is certain is that, after the months-long journey from Earth to Mars, and in particular the extremely dusty environment where it landed, it was normally expected that the ship would receive an extremely thorough, top-to-bottom and inside-out inspection and a great deal of routine maintenance before its next blastoff.  This was not done.

The most obvious reason that it was not done, is that there was no time.  The distance between Earth and Mars varies greatly over the course of their orbits, and travel between them is dramatically easier if Earth lies more or less directly between Mars and the Sun, which occurs about every 26 (Earth) months (of course the trip itself took several months).  The ship had made the journey to Mars so as to land a few weeks before this minimum, and the order to evacuate was given by Helene after they had been there for 24 (Martian) days.  Spending several weeks doing the standard, full pre-launch procedure would have put them outside the window of opportunity to make the trip back.

This meant that a full, standard launch prep was not possible, but it does still leave the question of why a couple days were not spent doing a few of the most critical steps.  There are several answers.

First, they DID do a few of these steps, the ones that could be done in a matter of hours rather than days.  You have to draw the line somewhere (if you are not doing the full procedure).

Second, as Helene pointed out in response to questions about this from her crew before the launch, it was not as if this possibility had never been planned for.  The reason that they had launched during the very early part of the window, rather than in the exact middle when the distance travelled would be at its absolute minimum, was exactly this scenario.  The ship designers had known that a rapid turnaround might be required, if conditions on Mars called for a quick evacuation.  It was not the best case scenario, but they had no time to wait for the best case, and it was not as if this was doing something the ship's designers had never considered.  In case it was required, Helene had the authority to authorize a rapid turnaround and return to Earth, balancing the risk against the risk of staying on Mars until the next launch window in a couple (Earth) years.

Third, and this was never spoken out loud, if they took more time to inspect and do maintenance, and anything was found that was worrisome (even if not actually a dealbreaker), there would be no time to fix it before the launch window ended, anyway.  All it would do was sow doubt and insecurity, given that Helene was determined to launch in any case.  She was not, after failing to sedate the Colonists, going to go back and tell them that she was staying for another two years.

The ship did not, technically speaking, ever launch; it attempted to launch.  Instead of the ignited fuel pushing out the bottom, giving the ship a fiery kick upwards, it spurted flame from several points on the side, and the spaceship crumpled to the ground, no longer a ship at all but rather a jagged jail cell for the inhabitants, on fire and leaking air fast, with electrical hazards and many other risks between the survivors and any hope of escape.

Theodore was heard to say, some time after he had been rescued, that what most impressed him about Helene's response to the explosion was that there was so little response.  She looked slightly annoyed, as if this was one more of many petty annoyances to be overcome, rather than a life-threatening, perhaps life-ending disaster for all aboard the ship.  Even after the ship fell heavily onto its side, and it was clear that everything had gone nearly as wrong as it could go, she was calm, and focused.

Unknown to Theodore, one reason Helene was calm, is that she was experiencing something akin to the feeling one has, not upon immediately losing a long-held dream, but somewhat later, when depression has replaced frustration or surprise or anger.  Helene had known, since she had gotten on the rover for its last trip to the ship launch site (nearly empty, as there were no Colonists to take with them, either conscious or sedated), that she had failed.  She had come to Mars to rescue the Orphans, and she had (with the exception of Harper) failed.

There is a certain skill to failing, that is underappreciated, and can only be acquired by practice.  That is the ability, after something unacceptably bad has happened, of resetting your expectations, so that you can once again choose between relatively better and worse alternatives.  If you cannot do this, then all of the alternatives before you seem equally, infinitely unacceptable, and the difference between them seems unimportant.  It takes an effort of will, and also some practice that can only be acquired by having things not work out for you, to change your mind, and stop thinking of all of the alternatives still left before you as equally unacceptable.  The alternative you wanted, is now out of reach, irretrievably, but it is difficult to stop thinking about it.  If you are going to choose intelligently though, between (for example) returning to Earth in defeat, or dying on Mars in a disastrous attempt at launch, one has to stop thinking about the prospect of returning to Earth as a hero, having saved several dozen abandoned orphans from a lifetime of poverty and isolation, and start thinking about what to do now that the original goal is impossible.

Returning to Earth alive would have been preferable for Helene, even if events had (she could not help realizing) in fact proven that she was not wiser than the Colonists, that using the ship to return to Earth was not safer than staying on Mars and repairing the fish farm, and everyone who followed her orders was worse off for it.  It would have been preferable to Helene, if she could have gotten herself to realize that even after failure, one must choose what to do now, from the options available.  Even if, since they were alive and still on Mars, the Colonists would be able to disparage her and her leadership in transmissions to Command back on Earth, while she stayed for two years on Mars and waited for a ship to come rescue her instead.  She could have perhaps even swallowed her pride, admitted her mistakes, and repaired relations with at least some of the Martians.  It was far short of the goal she had set herself, but it was better than dying in defeat and disrepute.

But Helene had almost never experienced failure in her life, and she therefore had no skill at dealing with it.  She had, of course, encountered difficulties, and had to adjust her strategy to get to a goal, or attempt multiple times to achieve it.  But she had never, in anything which she had truly set her mind to, been totally and irretrievably frustrated and defeated, and in the moment, she had no ability to see the difference between dying now, or any other outcome.  Except that dying now was quicker, soon over with.

Harper had been miserable, inconsolable, glassy-eyed and quiet during the entire trip to the ship.  Ophelia, who had been holding her hand since she left the Colony, might have been better able to respond to the rapidly-changing situation, if she had not been simultaneously hit with her new friend's miserable state, and her mentor's stony-faced air of dejection.  Most of the crew did not see anything amiss, as Helene was normally not prone to expressive outburts, but Ophelia knew her better, and she could tell that Helene was mentally thrown off balance in a way that Ophelia had never seen her before.

They had only a few seconds to move, to make their way towards the dock where the rover still sat, albeit on its side.  Helene stood, turned to look at her traumatized, disoriented and terrified crew, and shouted the commands.  

"To the rover bay, now!  Move, quickly!"

About half of them were able to comply.

Some were already unconscious or dead by this point, as the contents of the ship had been violently thrown about and in some cases broken; a few were bleeding to death from the shrapnel of the explosion.  Every spaceship launch was, in some sense, a controlled explosion; this one had not been controlled.  The ship had used nearly all the fuel intended for the months-long flight, in seconds.

Theodore reached down beneath his chair, grabbed the bubble-helmet that he would soon need to keep breathing, and moved with it towards the hatch that led to the rover bay.  As he was about to exit, he saw Loki, in his carrier, looking frightened and upset, and simultaneously angry.  Without giving the matter much thought, Theodore grabbed the carrier and took it with him.

Helene stayed until the end, trying to get more people up and moving.  For the most part, this was not possible.  If she had more time, she could have gotten Ophelia and Harper moving, but they were seated near the back, and in the dark and smoky atmosphere of the wrecked ship Helene could not see who was still alive and savable, versus who was dead or unconscious.  Therefore, she did not ever have the chance to get to where Ophelia and Harper were.  Ophelia, for her part, in addition to being stunned for a few seconds by the shock, was unwilling or unable to leave without her mentor.

It should also be said, that Helene was uninterested in the idea of getting herself out.  She spent precious seconds checking on clearly unsavable crewmembers, instead of moving towards the back, to reach where the rover bay doors were.  She did not actually think consciously anything like, "I hope it's over soon", but that was more or less the truth.  She concentrated on saving who she could save, but was (perhaps subconsciously) averse to actually moving very close to the exit herself, because that way lay a lifetime of regret and ignominy, and she could not see a reason to reach for that.

Harper, for her part, was both emotionally drained and also suffering from sensory overload, unable to comprehend what was going on or what she should do.  If they had been given even a few more seconds, Ophelia would have recovered from shock and dismay, and pulled Harper with her towards the rover bay.  As it was, they did not have enough time, although Ophelia never let go of Harper's hand, and would never have thought of leaving her.

The end, when it came in a second fiery blast, was mercifully quick.