As other answers have pointed out, the key to a system that utilizes renewable energy sources with an energy storage backup envisioned by the question is the ability to store large amounts of energy that are also readily accessible. Pumped hydro is the classical approach which has been in use for more than a century .
However, it is often difficult to find suitable sites for pumped hydro facilities, as they require suitable sources of water as well as easily exploitable differences in elevation. In addition, proposed pumped hydro projects are not infrequently challenged on environmental grounds. For these reasons, and with the advancement of technology, large-scale energy storage using electrochemical batteries is currently being pioneered , in particular in California which has few sites suitable for pumped hydro storage.
Battery storage systems in California are typically designed with a capacity that allows them to store four hours worth of energy at their rated power, so these are short-term storage solutions. As of summer of 2023:
Representing the largest concentration of lithium-ion battery storage on any grid in the world, the growing storage capacity – we reached 5,600 MW as of July 1 – is critical in decarbonizing the bulk power system and to our ability to keep the power flowing as California transitions to a carbon-free system.
At present, the largest single battery-based storage facility in California is the Moss Landing Energy Storage Facility, which is located on the site of a former natural gas powered power plant. This has the obvious advantage that much of the needed electrical infrastructure was already in place. The battery storage was installed in three phases, with the project completed by the summer of 2023:
An additional 350MW output and 1,400MWh energy capacity has been added to the plant, bringing it to a total 750MW/3,000MWh. This comes after the 300MW/1,200MWh Phase I was completed in 2020, followed by the addition of another 100MW/400MWh in Phase II the following year.
For comparison, the largest pumped hydro power plant in the United States, Bath County Pumped Storage Station, provides peak power of 3,000MW and 24,000 MWh of energy storage, so four times the power and eight times the storage capacity of the Moss Landing Energy Storage Facility.
Because of geography California utilizes relatively little wind energy but plenty of solar energy. The batteries charge up during the day when there is often a surplus of electricity produced from solar energy and discharge in the evening when the sun starts to set and usage often peaks, especially during the hot summer months.
The dashboard of the California Independent System Operator (CAISO) provides handy graphs that show how that works in practice. The battery storage facilities discharge in a staggered fashion depending on electricity market conditions, collectively covering about six hours. From casual observation, the daily peak discharge rate during the second half of 2023 seems to be around 3,000MW to 4,500MW, representing around 10% of California's electricity demand at that time.
Apart from occasional mishaps in the form of fires, large battery storage plants for short-term storage appear to be a solved engineering problem. My understanding is that the biggest remaining issue is the relatively high cost of battery storage, which exceeds the generation cost of electricity from solar and wind by factors. For example, for 2027 EIA projections are for a levelized cost of electricity (LCOE) for solar of \$36/MWh and onshore wind of \$38/MWh, but levelized cost of storage (LCOS) for batteries of \$125/MWh.
 The (to my knowledge) very first pumped hydro power station in the world was a 2.5MW facility in Switzerland that became operational in 1909, then called "Zentrale C" of the Schaffhausen power plant complex, but now usually referred to as Engeweiher power station. See: "Die Erweiterungsbauten des Elektrizitätswerks der Stadt Schaffhausen",
Elektrische Kraftbetriebe und Bahnen, Vol. 8, No. 21, June 24, 1910,
 According to the U.S. Energy Information Administration: "Before 2020, the largest U.S. battery storage project was 40 MW."