To "take up" space means to fill the space. As a result, other people or things can't use the space.
Why don't you get rid of that stuff? It's just taking up space.
Move over! You're taking up the whole bed.
As you can see, the phrase "take up the whole ___" is a common phrase.
Computers back then used to take up a whole room and had less processing power than today's cheapest cell phones.