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I have read questions and answers on measurable sets and from that I think this should be true. i.e.

Can we say that a set $S$ containing countable elements $S_i$ is measurable ? Further can we also say that we could define events on all subsets of $S$ while satisfying probability axioms.

Please correct me if I am using the language loosely or if I missed anything.

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Not necessarily. A set's cardinality is not relevant to its measurability. What sets are considered measurable are determined by the associated $\sigma$-algebra. It can be as coarse as $\{\varnothing, \Omega\}$ or as fine as $\mathcal{P}(\Omega)$, without that added bit of structure you do not even have a measurable space, which is an ordered pair, $(\Omega, \Sigma)$. Without $\Sigma$ there's not even a notion of "measurable" and it doesn't reference cardinality anywhere in there.

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    ok, can we define a sigma algebra on S ? – 2017-02-06
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    @hakunamatata yeah sure, go for it. There are tons and tons of ways to do this, pick your favorite. I mentioned two that are always there in my answer, the power set and the trivial $\sigma$-algebra. – 2017-02-06
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    ok, I understand why it doesn't make sense to talk about measurability of a set without a sigma algebra. Thanks ! – 2017-02-06
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    @hakunamatata glad to hear it! Cheers! – 2017-02-06