This is a slight elaboration on David Mitra's answer, but is too long to fit in a comment box.
To understand this question and how to solve it, you have to ask yourself what it means for the measure of the set where $f(x) \neq g(x)$ to be small (i.e. $< \epsilon$).
The smallest possible set is the empty set; to say that the set where $f(x) \neq g(x)$ is empty is to say that $f$ and $g$ are equal.
Now we can't necessarily take $f$ and $g$ to be equal, because $f$ may not be bounded, but we want $g$ to be bounded.
So instead, we are going to allow $g$ to differ from $f$, but only on a small set (one of measure $< \epsilon$).
Intuitively speaking, the "closer" $g$ is to the original function $f$, the smaller the set on which they differ will be.
So we want to find a way to change $f$ as little as possible while making it bounded. How can we do this?
Well, just choose a threshhold $n$, and define $f(x) = g(x)$ if $f(x) \leq n$, and $g(x) = 0$ otherwise. (The particular choice of $0$ here is not important, any number of absolute value $\leq n$ would do.)
Now $g(x)$ and $f(x)$ coincide unless $|f(x)|$ is too big.
So the problem is reduced to showing that the set of points where $|f(x)|$ is too big (i.e. where |f(x)| > n), which is the set of points where $f$ and $g$ differ, is small.
Now you have to use something: after all, if instead of a function on $[0,1]$, we had $f(x) = x$ on the whole real line $\mathbb R$, then the set of points where $|f(x)| > n$ (which is now just the set $(-\infty,-n) \cup (n,\infty)$) has infinite measure, and so is not small at all.
This is what David Mitra shows in his answer: because the total measure of $[0,1]$ is finite, and because $f(x)$ takes finite values almost everywhere, the set of points whre $|f(x)|> n$ has arbitrarily small measure, if we take $n$ large enough. QED
Some final remarks: from your comments on David Mitra's answer, I get the impression that you are thinking about this question in a very formal way. I would recommend that you practice translating formulaic expressions into more intuitive terms, i.e. try to read $\{x \, | \, f(x) \neq g(x)\}$ as "the set where $f$ and $g$ differ", and try to read $\mu(X) < \epsilon$ as "the set $X$ is small". Then you will have more chance of understanding what is really involved in a question, and hence have a better chance of answering it.