Part of the problem is that methods sections in contemporary journals do not provide enough information for exact replication, and in the most egregious cases let authors write stuff like "cultured cells prepared according to prevailing standards".
That means there are important validation/verification steps left out of the whole process. Sure, it's impossible to give every detail, and naturally there's always a time constraint, but if there's a hypothesis of action it needs to be verified. (Again easier said than done.)
This is how we write pen testing reports at work. A pen testing report written that way ~20 years ago is one of the things that got me interested in pen testing. But I apply it to all of my technical writing.
If lack of reproducibility in science is as big a problem as it seems to be, maybe journals should impose a randomized "buddy system" where getting a paper published is conditional on agreeing to repeat at least 2-3 other experiments performed by peers. Have 3 peer researchers/labs repeat the work. If at least 2/3 are successful, publish the paper. If not, the original researchers can revise their instructions once or twice to account for things the peers did differently because the original instructions didn't discuss them.
Hopefully needing to depend on the other organizations for future peer review would be sufficient to keel everyone honest, but maybe throw in a secret "we know this is reproducible" and a secret "we know this is not reproducible" set of instructions every once in awhile and ban organizations from the journal if they fail more than a few of those.
For corner cases that require something truly impractical for even a peer to reproduce independently ("our equipment included the Large Hadron Collider and a neutral particle beam cannon in geosynchronous orbit"), the researchers that want to publish can supervise the peers as the peers reproduce the work using the original equipment.
This would obviously be costly up front, but I think it would be less costly in the big picture than thousands of scientists basing decades of research on a single inaccurate paper.
I also think that forcing different teams to work together might help build a collaborative culture instead of the hostile one that's described elsewhere in this discussion, but maybe I'm overly optimistic.
That already happened in CS, which inherited slow and thorough journals from mathematics. Because peer review was taking too long, people elevated abstracts in conference proceedings to the status of papers. The idea was that you submitted an extended abstract to a conference with limited peer review. After receiving feedback, you would write the actual paper and submit it to a journal for proper peer review. But because the work was already published in conference proceedings, people often didn't bother with the full paper.
In some countries, administrators resisted this and only considered journal papers real publications. Those administrators were universally reviled by the CS community. Over time, most of them budged and started accepting conference papers as merits. And so CS became a field with lower than average standards for peer review.
For my masters' research I spent 6 years refining a super niche technique until I was able to reproduce my own work.