The always entertaining and informative Vsauce explains that the entire Internet's data only weighs about 50 grams. Go figure.
OK, technically, it takes millions of servers — an estimated 75-100 million servers — to store the Internet so you can count the weight of all of those servers as the weight it takes to store the Internet, but it's not the weight of the actual data.
But when it comes to how much the estimated 5 million terabytes of data that is the Internet actually weighs, it really weighs way less — about 50 grams — that's 50 grams of electrons in motion — the weight it takes to serve the Internet.
Getting a little nerdier, Vsauce says that all of the energy contained in the actual information — every single picture, video, email, website etc. collectively weights 0.2 millionths of an ounce. Putting that in comparison, that's an amount of mass in the size of the smallest possible grain of sand. Pretty cool to think about data fitting into a grain of sand, right?
Yeah, I think I need to go outside and mull this whole Internet thing over. It's so small, but it's so big.