How badly does failing a core class and retaking it hurt your chances at grad school programs?
At any stage in your career, people who will evaluate whether to give you an opportunity will (or at least should) be looking at multiple aspects of your CV. For each &quot;weak&quot; aspect there should be a &quot;strong&quot; aspect that offsets it. Provided this is true, you should become eligible for consideration. After that, it is often a matter of luck and the competitive nature of the opportunity.
Having said, that, there are places and systems which I personally know of that will automatically reject applications that have a single blemish. These places and systems are making a mistake (in my opinion). In any case, perhaps you would not want to be in a place that does not believe in giving people a second chance.
In summary, I think there are a number of good places which would give you the opportunity if you can counter this particular failure by something---which could even be a good grade in the same course on your second attempt.