Engineers routinely talk about the difference between what is possible in theory, what can be demonstrated to be possible in the laboratory, what can be done in practice, and what can be done economically. There are lots of inventions that have been built, demonstrated, and really do work, but that are far too expensive in one way or another for anyone to use for any practical purpose.
Like, auto manufacturers have built prototype cars that run on fuel cells, meaning that basically the fuel is water. Sounds really great: if we built a fleet of cars that run on water, think how much we could save on gasoline, not to mention cutting pollution, dependency on foreign sources, etc. But there's one small catch: the raw materials for each car cost about $200,000, not even considering what it costs to actually assemble the vehicle.
My point being: If tomorrow we all woke up and discovered there was some really compelling reason why we had to go to Mars, if the future of all life on Earth depended on it, the basic technology to make such a trip exists. Engineers would have to work out the details of the design, but I'm 90+% confidant that, if a hundred billion dollars was devoted to the project, and if all the bureaucracy and environmental regulations and political considerations and all were waived or ignored and suitably capable people pulled from whatever they were doing before, we could have the vehicle built and launched in a year or two.
The problem is what would give us the motivation to do all that. Given present technology, the cost of transporting one person to Mars would surely be in the billions of dollars. It is difficult to imagine anything they might find there that would be economically productive. Whatever resources exist on Mars, we could surely obtain the same resources here on Earth, or find some way to accomplish the same goal with different resources, for far less money. Despite all the problems here on Earth, it would almost certainly be more productive to devote resources to solving the problem than to escaping it by going to Mars.
Consider why Europeans explored and colonized the New World. (a) Looking for a more efficient route to India. (b) Access to gold and other resources. (c) Scientific inquiry and adventure. (d) Political prestige. (e) Escape from religious persecution.
With our present technology, economic motives like (a) and (b) are unlikely to be practical. (c) is possible. At present a manned expedition is so expensive that no one is prepared to pay the bill just for the shear joy of discovery, but it could happen. (d) is possible but the political situation doesn't presently exist. (e) is too expensive for any dissident religious group to be able to afford it at present.
All that said, my conclusion is that no one is likely to send a manned expedition to Mars until there is some technological advance that dramatically reduces the cost, OR until enough people think the scientific or political benefit is worth the cost. It's difficult to predict when either of those things will happen. I'd be surprised if there's a manned expedition in less than decades, and a true colony would be further away still.
Of course for a fiction story, you can invent circumstances to make it all happen. You can just write, "... and then Dr Jones invented the new Fubar drive that revolutionized interplanetary travel ..." or "After the rise of the Islamic Caliphate, the United States and the Caliphate became involved in a space race to see who could reach Mars first. Everyone knew that whoever won would be viewed as the technological leader, and other nations would quickly join their alliance ..."
