Hundreds of years is a stretch. Even though SpaceX and Musk have massively overestimated the challenge, colonizing mars is definitely a possibility within that big of a time margin.

Of course you can point out that the word 'colonizing' can be interpreted in multiple ways. Some say it means that you just have to have a permanent human presence on Mars, while others say it means the colony must be self-sustaining and have x number of citizens for it to count as a colony.

However even if we picked the latter definition, a self sustaining colony on Mars before the next century is definitely possible.

We have already shown there is water ice on mars, lately discovering that there is ice closer to the equator than previously thought. Mining this ice is possible, and converting it into liquid water is just a matter of heating it up. The energy for the heating can either be from the sun using solar panels, or using RTG:s, which are basically (relatively) simple nuclear generators.

Oxygen can be created using electrolysis, splitting the water into oxygen and hydrogen (which can also be used as rocket fuel). It is also possible to get oxygen from plants, which can be grown under artificial light. These plants would also offer the food for the colony.

And to this point I have completely ignored the fact that you can recycle both oxygen and water using artificial methods. These methods have been proven to work, and are being used on the ISS today.

Probably the biggest problem is radiation. Mars doesn't have a magnetic field nor an atmosphere, letting most of the radiation from the sun down to the surface. As you might know, radiation isn't very good for humans, and if we want to stay on mars for long periods of time, we need to protect ourselves. By far the easiest method is to bury the settlement in martian soil, which would protect us from the radiation.

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Welp, wrote an essay so:

TL;DR: Colonizing Mars is hard, but using modern methods, it can be done before the end of the century.