You don't necessarily need fossil fuels for steel production. It is also nothing impossible in capturing CO2 from cement making - it's mainly centralized production and much simpler problem than transportation.
You do need a carbon source for steel production. It’s integral to steel. You don’t need coal per se, but you do need to burn carbon. Every technique so far will release some CO2 as a waste result.
The steel industry is one of the largest emitters of CO2, contributing 6%–7% of global greenhouse gas emissions.
That said, there's a difference between sourcing that carbon from fossil fuels, carbon "new" to the current land|sea|atmosphere cycle having been drawn up from where it was sequestered millions of years past, and carbon that is already part of the surface dynamic.
Major steel players, those in the billion tonne per annum mining and processing chain, are already going hard at replacing current steel making with several alternatives and have already built pilot plants to trial low | zero "new carbon" production techniques.
The carbon needed for steel as an alloying element is a small fraction of the carbon needed for reduction of iron ore. The latter is replaceable with renewable energy (hydrogen or direct electrolytic reduction).
Also, note that 70% of steel production in the US isn't from ore at all, but is from scrap metal. Most of the steel used in renewable energy infrastructure will not be consumed, but will be recycled.
On the other hand, steelmaking also uses limestone as flux, to remove impurities from molten steel. When heated on the steel this drives off CO2. Calcium oxide could be used instead, but then that has to be sourced without CO2 emission, just as in cement manufacture.
I think there are at least two ways to make lime without CO2 emission. The first is normal limestone calcination, but with CO2 capture. The other is to use calcium silicate. This means dissolution of the silicate with hydrochloric acid, separation of silica, then high temperature reaction of calcium chloride with steam to produce lime and hydrogen chloride. I understand there's a company trying to commercialize this latter process.
You can eliminate the vast majority of emissions by using hydrogen. Then the waste will be water instead of CO2. This is no longer hypothetical. Pilot plants have been successful enough that at least SSAB is investing billions into hydrogen-based steel production.