Your best bet with storing important documents is a safe deposit box.
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U.S. citizens who arrive at an Italian Port of Entry without a valid travel document – including passports that have been previously reported lost or stolen – will be denied admittance into Italy and returned to their point of origin. This regulation is strictly enforced in Italy.
Moreover, when you get to Europe, you will still need your passport as it is your identification document. You can use it to book hotels, get local SIM cards, prove your identity to the authorities, and move from one region to another.