Q:

What does the following question(bold) mean?

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?

A:

are they spaces through which the forces of big capital are further extending into our everyday lives?

You don't mention any specific part that is giving you difficulty, or how much of it is clear, so I'll do my best to break it down generally.
Let's start with the subject of the question.  "The forces of big capital" is effectively one big noun.  Capital in this context means money, like in the word capitalism.  It doesn't refer to a political capital, like Washington DC is that capital of the United States.  Big Capital is large corporations that have a lot of money.  The forces of big capital refers to how big corporations like Facebook have a lot of money, and that means they can effect our lives.  Smaller companies can push with a little force.  Big companies can push with a lot of force.
So, those forces are "further extending into our everyday lives."  Big corporations already have some impact on our lives.  But they are trying to have even more control over our lives.
Social media sites like Twitter are "spaces."  Space is a metaphor here -- it doesn't mean you have to get in a space ship and fly into outer space to post a Tweet on Twitter.
...  So, a very similar question that might be easier to pick apart:
"Are Web sites like Twitter and Facebook being used by wealthy and powerful companies to have more power over our lives?"

