It's probably different from person to person, but the internet doesn't make me feel small. It makes me feel almost limitless because there's an incredibly massive wealth of information and communication that's accessible to me, just like everyone else.
Even without the internet, just knowing our place in the universe is enough.
Neil deGrasse Tyson put it best: “I look up at the night sky, and I know that, yes, we are part of this Universe, we are in this Universe, but perhaps more important than both of those facts is that the Universe is in us. When I reflect on that fact, I look up—many people feel small, because they’re small and the Universe is big, but I feel big, because my atoms came from those stars.”