Short answer: Self-reference isn't the problem; reference is the problem.
Longer answer: Let's examine this statement S1. What does it mean?
S1 means "S2 is false and S3 is false and S4 is false and..."
OK, let's expand that out some more, using the definition of S2:
"(S3 is true or S4 is true or S5 is true or...) and S3 is false and S4 is false and..."
OK, we can then start expanding this out using the definition of S3:
"((S4 is false and S5 is false and...) or S4 is true or S5 is true...) and (S4 is true or S5 is true or....) and S4 is false and S5 is false..."
I think you can see, this expansion will never terminate. It is impossible to fully write down what S1 actually means.
In modern mathematics, we require that it be possible to write down a statement in full, without referring to other statements -- statements should have meanings by themselves. Technically speaking, statements are not allowed to refer to each other at all. Now of course, if I have statements like
T3: 1>0 and 2>1
Then I can sum up T3 as "T1 is true and T2 is true"; but "T1 is true and T2 is true" is just a human-readable summary; it's not the actual formal statement. Formal statements, as I said above, must stand on their own and are not allowed to refer to each other.
This is why Yablo's paradox is not a problem in modern mathematics.
(Note, by the way, that Russell's paradox does not use this sort of self-reference -- or reference of any sort. The liar paradox can't even be formulated in the modern framework of first-order logic; Russell's paradox can be, and just requires the appropriate axioms to appear. It does definitely have a sort of self-referential quality to it, but it's not the same thing.)