The question seems a bit absurd, since we don’t think of the internet as a physical entity. However, we know that the information that comprises the internet is stored as electrons, and Einstein told us that electrons have mass. Therefore, neglecting the servers that the internet is stored on, the internet has a specific mass. A random email, say, is roughly 50kb, and that amounts to about 8 billion electrons. That may sound like a lot, but that amounts to only two ten thousandths of a quadrillionth of an ounce.
If an email is that light, how much does the whole shebang weigh? As you might suspect, scientists disagree over the best way to calculate the mass. Richard Seitz, on his blog Adamant, claims that the web is about 2 ounces - or roughly the mass of a plump strawberry. Others, however, say his estimate is too high; the true value may be much smaller, closer to the mass of a grain of salt!
That just goes to show how a massive amount of energy need not weigh very much. And that, fellow short people, is why no one is allowed to make fun of us again, according to science!