To "take up" space means to use that space. But "take up" is a slightly negative expression. It's used when space is important, and you don't want the space to be used:
We used to have a piano in the living room, but it just took up a lot of space and no one ever played it.
You can use the word "space" like in this example. Or you can name the space, like in the example at the top or in the following:
That old car is taking up the whole garage.