Any of the high-temperature reactor designs (sodium, molten salt, TRISO, etc.) can be combined with thermal energy storage.
Light water reactors only get up to 300°C, which is barely hot enough to spin a turbine. At >600°C, you can heat an intermediate fluid, lose some energy, and still spin a turbine.
So the idea is that there are chambers of molten salt (or similar) that can be heated when a spinning turbine is not needed, and it'll stay hot enough to spin one later? Any idea how long it stays hot? What happens when they're all already heated?
12-24 hours of molten salt storage should be enough to generate power all day, and sell it at times when electricity is most expensive. There's not much point storing for multiple days, when the reactor itself behaves like long-duration storage.
If your reactor is producing energy with no buyers, and your thermal storage is full, then you probably should've built it somewhere else, but in that case you can just power down the reactor.
Light water reactors only get up to 300°C, which is barely hot enough to spin a turbine. At >600°C, you can heat an intermediate fluid, lose some energy, and still spin a turbine.