The Dragon is roomy enough for a few-day trip to the ISS, but it might become pretty cramped after a couple weeks for two people, let alone the 500 days it will take for the trip to Mars and back. The trip will be spartan. One of the biggest dangers of the trip might be psychological — humans, like rats, can become aggressive, and even murderous when confined in too-close quarters for too long. And packing enough food and water for that many months won’t leave a lot of volume for stretching out, or basic privacy.
And even if they manage to survive the trip without killing each other, radiation in deep space will be a major issue. Most NASA plans for Mars missions assume the use of large amounts of water for shielding, but there isn’t enough volume or mass to have that in the Dragon. Other health problems will arise from the long period spent in weightlessness, which deconditions the cardiovascular system and reduces bone density, and even affects eyesight. But for this mission, that won’t be a problem until the return to earth, since there are no plans to descend into the gravity well of Mars or operate on the alien world.
Some of these problems — the lack of habitable volume the radiation issues, for example — could be alleviated with a larger vehicle, such as a Bigelow BA 330 module. Coupling the Dragon with such a facility would make the trip relatively luxurious, but also much more costly in terms of required mass delivered to orbit, demanding multiple Falcon Heavy flights, rather than a single one. Between the extra launches and the cost of the Bigelow module, it would take the mission costs from less than half a billion to closer to a full billion. On the other hand, it would allow more people to go in more comfort, so perhaps the money could be raised with additional ticket sales.
One thing is certain — this is not a mission that NASA, in its institutional risk aversion (encouraged by Congress) could or would attempt, particularly since it would have the appearance of a stunt, with no scientific benefit. If it occurs, such boldness will be left to the private sector. And that’s perhaps not such a bad thing. In any event, we’ll find out when they make their formal announcement early next month, a week from Sunday.