Finally, at a grander level, are ‘free’ social networking sites user-owned, or are they spaces through which the forces of big capital are further extending into our everyday lives?
As a 50-year native U.S. speaker, I will attempt to interpret this without bias, starting with defining the key phrases so we are "on the same page" with a common understanding.
First (and I am not correcting you here), I would capitalize "Big Capital" to make it more clear that this phrase is a particular concept. Big Capital apparently refers to the largest forces of capitalism in the world, in the same way that Big Tobacco refers to the largest, most influential tobacco companies, and Big Pharma refers to the largest, most influential medicinal drug manufacturers.
I'm not sure if "spaces" is already clear, but to be sure, I interpret that here as stating that the social networking sites are simply a "space" themselves. This could perhaps be thought of as an empty vessel, like an empty vase or drinking glass, or as something open at two ends, a "conduit," like a tunnel through which things can travel. It is a space waiting to be filled with something, or for something to go through it.
I'm going to assume, for now, that the rest of the phrases used in the highlight are correctly understood.
With those interpretations, I would interpret the highlighted sentence in this manner (saying nothing about whether or not I agree with its meaning):
... or are apparently-free social networking sites actually tunnels through which the richest companies, primarily focused on making more profits, are reaching beyond traditional print advertisements, billboards, and TV commercials (which one can avoid seeing) and reaching us through other common things we all do every day (implying that it is harder to avoid seeing what the companies want us to see, if we want to use these sites)?
Does that help?