Here is one way to do it.
Get stock data:
ticker = "AAPL";
dateRange = {{2020, 1, 1}, Today};
originalData = FinancialData[ticker, "OHLCV", dateRange] // Normal // QuantityMagnitude;
Create a file name:
fileName = StringJoin[ticker, "_", DateString[Today, {"Year", "Month", "Day"}]]
Put in file:
Put[originalData, fileName]
Read in and check:
newData = Get[fileName];
newData == originalData
You can gather up the code in a function to create files for all the stocks and date ranges you'd like to store.