How can one construct or prove the existence of an example of a nowhere dense set in $[0,1]^2$ with Lebesgue measure $1$ using real analysis? Osgood gives an example in http://www.maths.ed.ac.uk/~aar/papers/osgood.pdf (also see pg 135 of http://www.kryakin.org/am2/_Olmsted.pdf) but I am not familiar with Jordan curves nor the technique being used. This is from an old qual, and I would like to know if it can be solved using real analysis.
If I cannot construct a set of measure 1, how can I construct a set of measure $1-\varepsilon$ if such sets exists?
The construction of fat Cantor sets (you might google "Smith-Volterra-Cantor set") shows us how to build a closed subset of $[0,1]$ with empty interior whose measure is arbitrarily close to $1$. Taking a union of a sequence of Cantor sets whose measures approach $1$, we get (by the Baire category theorem) a set with empty interior of full measure. But this set is not nowhere dense; in fact it will be dense.
Everything I said here extends straightforwardly to higher dimensions.