CH has not been "settled" (and there are obstacles to settling it) in any of the following senses:
$\quad 1.$ Finding a compellingly natural extension of standard set theory (more natural than ZF+CH) that decides CH, i.e., proves CH or proves its negation.
Here the main approach is blocked, because large cardinal axioms don't directly decide CH.
$\quad 2.$ Finding compelling arguments for replacing set theory, wherever it is used (e.g., as a foundation or formalization scheme), with set-theory-plus-CH.
This approach is blocked by the lack of "material consequences" of CH. For example, the set of true first-order sentences of arithmetic is not affected by assuming CH, so there would be no concrete statement such as the Twin Prime Conjecture that could be proved only with the use of CH. For similar reasons, it is unlikely that there exists a proof of any concrete statement that is much shorter or easier with CH than without it.
$\quad 3.$ Finding a compellingly natural alternative to standard axiomatic set theory (one whose theorems are not a subset or superset of the theorems in ZFC, and which comes to be preferred over ZFC) that can formulate and decide CH.
This development would be a lot more significant than deciding CH, and would presumably affect a large number of other questions. So to the extent that this possibility is relevant it should be discussed directly, and CH itself is irrelevant. More on this argument in the earlier thread: Knuth's intuition that Goldbach might be unprovable
(The same comments also apply to the negation of the Continuum Hypothesis; the above is phrased in terms of CH only to avoid clunky qualifiers in the sentences.)