...build a gazillion batteries to hold an entire continent power grid up for 28 hours straight.
This is a rediculous requirement.
28 hours of *total contintenal* battery power? Absolutely not required, unless we're somehow worried about a scenario where the wind doesn't blow, and the sun is eclipsed, and there's no water flowing down power dams, more than a day straight across the entirety of the continent. In which case, by my estimation, problems with available electricity will be rather low on the list, as the Vogon fleet blocking the sun will be clearing us away for their hyperspace bypass anyway...
"Renewable" energy includes solar, wind, and hydropower. Hydro's convenient, if massive to build; once the dam is complete, one tends to expect it can produce at- or- near capacity unless the water level drops. The Hoover dam's a good example of that changing -- over the course of years, not days or hours -- but Niagara, as a counterexample, is being *expanded*. So there's a source of 24-hour power available, regardless of any short-range weather systems.
Solar is highly consistent. Every day, the sun comes up and shines; and, across the continent, large fractions of land receive direct open sunlight. Even in areas clouded on a particular day, solar still provides some fraction of the nominal load. Overbuilding resolves capacity requirements, and planning for overnight darkess is predictable.
Wind, also reasonably predictable. At the height of the commercial generators, and in locations where such generators are built, wind tends to blow most of the time, at a fairly steady pace, and all day. There will be times when the wind drops -- or goes too high -- but wind generation helps cover the solar overnight. Again, overbuilding - and building in multiple areas - tends to negate the low-generation windows for any specific area. If the wind stops in the southeast, it's probably blowing in the northeast... or southwest. Or both.
So, then -- what happens overnight on a still night? The grid! Power from the windy northeast can be redirected south to the storm-covered southeast, or west, or... and locallized battery installations can maintain power for a period, as well. This is exactly what already happens today, when a particular power plant is taken offline; the grid routes power from elsewhere.
28 hours of continent-wide battery capacity? Expensive and wasteful; the doomsday scenario where it could be 'needed' is irrelevantly unlikely.
~24 hours of *local* capacity, depending on the specific mix of readily available grid power sources? Might be reasonable.
12 hours? Fairly reasonable, for those overnight, still days. Though even there, the grid's likely to be able to compensate from somewhere, most of the time. And achievable using hyperlocal solutions, including local home batteries and vehicle-to-grid.
All of that, *and* reducing greenhouse gas emissions, while producing power at cheaper total cost. Yes, Coal and Gas plants will go away, eventually, at least for areas where the grid can compensate for localized changes in renewable power. Island nations and isolated communities, particularly in very remote areas like the far north, might need local, fossil-powered generation for a while longer, but continental grids like North America and Europe should certainly be capable of removing those power sources, given some time to continue building out renewable generation and storage capacity.