Is a manned mars mission plausible in the near term?

So, this was provoked by the Trump speech, which I admit is a silly source, but I was wondering, would it be actually possible in a reasonable timeframe given the technology we have?

I thought that because of the long distance, and the frequency of solar flares, any astronauts would be cooked by radiation long before they reached Mars, unless they could bring a substantial chunk of mass to hide behind. This implies that it should be impossible given the current cost of carrying heavy things into space.

Is my reasoning sound? If not, what gives?