What exactly was wrong with Steve Rogers that prevented him joining the army?

In the comics, Steve Rogers’ 4F condition was described in vague terms as “frailty,” and his visual depictions usually suggested mal-nourishment, which isn’t surprising given the poverty of his early life. 

The MCU expanded upon this, showing us his medical intake report:

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It’s a pretty comprehensive list – asthma (which cannot make the scarlet and rheumatic fevers or the sinusitis or chronic colds any easier to bear), a whole cluster of heart problems which were probably due to rheumatic fever having caused rheumatic heart disease, a generally poor immune system, and a family history of tuberculosis. 

And a lot of these conditions – especially asthma, heart disease, scarlet fever, rheumatic fever – are illnesses associated with poverty.