Unfortunately, all you can do is leave them on the floor. Yes, really, just leave them on the floor in your house until you want to sell. There are no consequences to leaving your turnips on the floor. If you really want to get into the stalk market, consider making one of your rooms a dedicated turnip cellar.
Just don’t forget to pick up all your turnips when you’re ready to sell! Turnips rot once the next Sunday rolls in.