The Question
Let $X$ be a set. Let $\mathcal{F}\subseteq P(X)$ be a $\sigma$-algebra. (Or, if it makes a difference, let $X$ be a topological space and $\mathcal{F}$ the Borel sets.) When can we guarantee the existence of and how can we construct a non-atomic probability measure $\mu$ on $(X,\mathcal{F})$? (In addition to being a measure, we ask that $\mu(X) = 1$ and that if $F\in\mathcal{F}$ and $\mu(F) \neq 0$, there exists a proper measurable subset $E\subsetneq F$ such that $0 < \mu(E) < \mu(F)$.)
Some remarks
The result of Sierpinski guarantees that $\mu$ actually takes on a continuum of different values. So necessarily $\mathcal{F}$ needs to be uncountable and hence $P(X)$ has to be uncountable. This puts a lower bound on how many elements there can be in $X$. (And trivially one sees that if $X$ is finite any probability measure must have atoms.)
I expect, however, that there may be other necessary conditions for the existence of a non-atomic probability measure.
On the sufficient side, the only result I am familiar with that explicitly constructs a measure is the various constructions of the Lebesgue measure. This construction makes use of the local structure of Euclidean space and hence also works for, say, topological manifolds. (Okay, there's also the ultrafilter construction for additive measures, but while the constructed measures are probability, they have a lot of atoms.)