This is a rather broad interpretation of "Cantor-type." The basic idea is the following (letting $Q_i$ be an enumeration of the countably many nontrivial open intervals with rational endpoints):
We have countably many "positive requirements" $P_i\equiv m(E\cap Q_i)>0$. The obvious way to meet such a requirement is to put an interval into $E$.
We have countably many "negative requirements" $N_i\equiv m(E\setminus Q_i)>0$. The obvious way to meet such a requirement is to throw an interval out of $E$.
The obvious issue is that the action we take to satisfy one requirement might "injure" another, e.g. putting $(0,1)$ into $E$ to make $E\cap (-3, 17)$ have positive measure obviously causes a problem for $E\setminus ({1\over 3}, {1\over 2})$. One solution is to allow injury but only to a very controlled extent; e.g. in the above example we might allow ourselves to put a very tiny subinterval of $({1\over 3}, {1\over 2})$ back into $E$. More abstractly, we'll want to keep track of some numerical parameter during our construction which tells us how much we're allowed to "alter" the set being built.
The details are, in my opinion at least, fun to work out for oneself, so I've spoilered the construction I have in mind:
We'll define a sequence of sets $E_i$ (= the stage-$i$ approximation to $E$), as well as a sequence of "stability parameters" $\epsilon_i$, as follows. We start with $E_0=\emptyset$ and $\epsilon_0={m(Q_0)\over 4}$. Next, having defined $E_i$ and $\epsilon_i$, we now face the pair of requirements $P_i$ and $N_i$. We pick a pair of disjoint open subintervals $I,J$ of $Q_i$ with $m(I)=m(J)=\epsilon_i$. Of course this requires $\epsilon_i$ to be "small" relative to $Q_i$; that will be true by induction, and it's with this in mind that we set $$\epsilon_{i+1}=\min\{{\epsilon_i\over 2^{i+2}}, {m(Q_{i+1})\over 2^{i+2}}\}.$$ More obviously, we also set $E_{i+1}=E_i\cup I\setminus J$. Finally, we take the "limsup" of the sequence of sets we've built so far: $$E=\{x:\forall m\exists n>m(x\in E_n)\}.$$ (The "liminf" would also work, the point is just that we need to account for oscillation of membership in the $E_i$s as $i\rightarrow\infty$.) Since the sequence $(\epsilon_i)_{i\in\mathbb{N}}$ shrinks sufficiently rapidly, and is always small relative to the interval being looked at at a given stage, it's easy to see that all requirements are satisfied.
