I think there's at least 2 very different related questions:
Is it possible to have a not-too-large-to-build colony on Mars be capable of surviving if contact to Earth is cut off?
Can you make a colony on Mars that's capable of (1) and is economically viable?
I think the answer to (1) is yes, probably with a lot fewer people than you might think, but the answer to (2) is likely no, regardless of number of people.
Robin Hanson, in other discussions has made a big deal of the importance of specialization and division of labor. While he's right that it's pretty important to our modern economy, another lesson of economics is that things usually have substitutes. We can use less division of labour, but it just costs more and you get less good results.
Another commenter brought up electronics. I agree that electronics would be hard to do without a lot of people. If we are talking about (1), raw survival, then I disagree that electronics is needed. For (1), the question is whether people can survive at all - we can assume that we have a colony full of very capable, hard-working people who work long hours to maintain a bare-survival standard of living. The minimum size of the colony is likely going to largely depend on one or a few products that they absolutely need and require some amount of division of labour to produce - possible examples might be spacesuits and equipment for power generation or mining. By contrast, something like farming or habitats for farming in might take up most of the economic effort, but not lower-bound the colony size so much since less division of labour is needed.
For (2) on the other hand, to have an economically viable colony you need people to be willing to come over from Earth. Earth has free air, cheap water, you can grow stuff outdoors. No amount of division of labour on Mars can make up for this advantage, since Earth already has a lot of division of labour in addition to being habitable. So you need a lot of imports from Earth per person to get a comparable standard of living on Mars, and I don't see enough plausible exports to sustain that large amount of imports.
Of course, if people are willing to sustain a large enough drop in material standard of living in order to live on Mars, it could be economically viable, but you'd need a decent number of people willing to have a really massive drop in standard of living. Are there enough such people? I doubt it, but would be happy to be proved wrong.