Yep, doable and already getting done in lots of places. Discussions like this usually devolve into alarmist what ifs and vague assertions about needing something called "base load" which is a surprising poorly defined notion. There are now several places in the world that regularly have hours/days/weeks of being exclusively powered by renewables. And it's fine. When that happens the cost is low. When it doesn't they pay more to import power from elsewhere or they switch on some peaker plants. Typically, without any outages or downtime.
Applying some system thinking is indeed key. If you look at each solution in isolation, they indeed each have issues but they are different issues. If you take them altogether, you end up with a resilient grid network with much less issues. Wind by itself has issues. But together with solar and some short term storage, it gets a lot more resilient. There are still some issues left when you do that because cold gloomy winter days with no wind are a thing and those conditions can last for days or weeks in some places. We don't (yet) have storage to bridge such gaps. Months is actually unusual but weeks is fairly common in places like Germany in the winter.
So, you can't obviously rely on that mix exclusively. Which is something nuclear proponents love to point out forgetting that turning nuclear plants on and off is really expensive and slow and generally not something that is done regularly. They kind of suck for backup power. Nuclear peaker plants are not a thing. Hence they like to talk about base load because that means leaving them on permanently to provide that base load.
The alarmist view to this is that without this base load we need to have enormous amounts of storage to survive these horrendous apocalyptic spells of gloomy days (i.e. winters). Exaggerating here but this goes to the core of what nuclear proponents advocate: yes it is stupendously expensive but we have to have it because we need the "base load". The fallacy in that argument is pretending that nuclear is the only option for this and ignoring the cost aspect. Also, nobody ever specifies how much of this base load is actually needed (in gwh). They just assume that we need lots of it.
The system thinking pragmatic real world solution to this is realizing that these gloomy conditions are typically localized, seasonal, predictable, etc. and that running some cables across the continent adds a lot of resilience. Like Scandinavian hydro power, or solar power imported from places like Spain or Morocco (both of which are a thing). Moving power around with cables means you can shape and shift demand around as well. Mostly we're not talking about 100% collapses in generation but supply and demand variations of more reasonable percentages.
And of course the reality is that we have all these legacy plants still providing much more base load than is actually needed right now. We don't actually need more of that right now. It's not an urgent problem (aside from getting rid of emissions). And they aren't going to be switched off overnight and will be around for quite some time. The mix is gradually shifting to more and more renewables, all sorts of storage solutions. It's going to asymptotically converge on 100%.