Well, let us remember a few things when making that comparison:
1) The Niña had a complement of 24. The two missions to Mars being talked about much these days have crews of 2 and 4. For social animals like humans, this is a major difference. And while Columbus' ships were small, they still had notably more room than a Dragon capsule - you could walk around on deck, if nothing else.
2) A 3 month trip on a small boat isn't fun. The Mars missions under discussion are more like 8 months one-way.
This is not to say it is impossible, but merely to be realistic about what we are asking.
There are logistical challenges, yes, but the general attitude is that they are insurmountable, which they are not. We assume that our technology is at the same level today, which may or may not be true with 60+ years of confirmed knowledge of extraterrestrial beings in our solar system, but even with no additional impetus to develop space technology any further than what is available today, we do have the technology to build a space ship capable of carrying 24 people. It's a matter of cost on why we don't, and I suspect that with a known alien species at the other end, we could come up with that funding over say, a 20 year development plan that could have begun in the 70's. I also suspect we would have built some semi-permanent moon base by now, allowing the larger ship to travel from our moon to one of Mars' moons, cutting the cost of running such a vessel considerably, since we'd only need to boost our ship into space once. A large part of the cost of any mission is the expense of lifting all the weight out of a planet's gravity well, but if that is defrayed over a larger number of missions, it does become economically feasible to trade luxury items and items of high scientific value. Martian spiders that can spin their silk in thicker strands at a higher tensile strength could potentially replace steel alloys in suspension bridge construction, for example, making a land bridge across the bering strait much more economical.
Even if we scaled it down to say, a ship designed for 6 or 8 people, you also don't account for the difference near-instantaneous communications would have for socialization purposes. Looking at similar areas of isolation, Antarctica has 70 research stations and a population of 1000-4000 people, depending on season. In the winter, assuming even distribution of the population among those stations, you're looking at an average of 14 people per station. Additionally, Sergei Krikalev managed to live 10 months aboard Mir during the dissolution of the Soviet Union, and that had a maximum crew of 3 people.
There is indeed a psychological component to the extreme isolation such a journey would entail, but we already have people who've been in similar situations in the past, and it isn't hard to arrange a dry-run of potential candidates to see how they'd react to living and working in small confines with the same crew of people. The reason this is even considered a barrier at present is because a manned mission currently can't bring home anything that a robotic probe couldn't do safer and cheaper. Having an entire world at the other end changes that entirely. Even if the lifeforms on our psuedo-Mars were completely different than anything on Earth, it would still give us another view of how life could have started, a different set of data to add to what we know of evolution, etc.
Edit: Replying to the post by Morrus, you are correct. Further research has proven my ignorance, the flat earth theory was largely abandoned around the turn of the millennium, even among the uneducated populace. I withdraw my incorrect assumption.
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