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Since portals aren't real, maybe we can start from what we see in game: Chell can hold it for long periods of time without problem, so that puts an upper weight barrier on it. Aperture must've figured out a way to fit portal technology into it, or found a way to stash its mass away. Or maybe it's heavier than average, but Chell is drugged to be able to hold it.
There's no power cord in-game, so its energy source must be self-contained, because I don't think it can get enough power by absorbing things in the air.
It's important to note that Chell has reinforced legs. That is important for the whole weight distribution thing. She can probably lift heavier things or at least become less fatigued carrying something for long periods of time.
I think you meant Aperture
thanks! I've fixed it.
Regarding a power source, we only use it within Aperture, so it might be powered through some kind of induction charging in the walls.
Or the power source is somewhere else and the gun gets its power with a cable running through a tiny portal.