It's worth noting that the majority of papers get rejected in peer review at least once. They get sent back for revision, to fix some errors of logic, statements that go beyond the data presented, or they get rejected as bad science, or they get rejected with a suggestion for a more suitable journal.
But peer reviewers never rerun the experiment. That's the next stage.
A journal article contains a careful description of the materials and methods used in the study. Other researchers take that description and attempt to replicate it. If it doesn't work, people get suspicious. (See also cold fusion.)
No one caught any error beforehand with the Korean stem cell paper because there's no doubt that it is possible to produce patient specific embryonic stem cells. The question is how.
I'd make one point on top of those raised by the bloggers above.
There is another layer of checks on top of peer review and scrutiny by other scientists. In America, most basic research is funded by the government. Non-partisan panels allocate funding, they provide ethical oversight.
I don't know what oversight the Korean government provides, but I'd wager the NIH and NSF provide more. And that's an argument for US government funding of stem cell research. Fund it, put reasonable limits on ethical practices, and let the research proceed under the brightest spotlight possible.