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Raspberries (and blackberries and all those similar) have a short shelf life which is why you rarely see them sold in stores. They might last a day or two but go bad fast (even with refrigeration.) Blueberries also go bad quickly but can last a bit longer. Yes, refrigerate them, but don't wait too long to eat.
One to Two Days. Fresh strawberries can go directly into the refrigerator, but will do just fine on the counter for a couple of days. Remove any bruised or otherwise marred berries and place the rest in a colander or open-weave basket to allow good airflow.