For a pure state of 8 qubits, the Hilbert space is $2^8$ dimensional. Dropping the normalization and phase information means you are left with the space $\mathbb{CP}^{2^8-1}$.
Unlike a single qubit which give the Bloch sphere $\mathbb{CP}^{2^1-1}$, this is too big to draw directly. Instead one usually draws simpler spaces that capture the essential features. This is thanks to the fact that these spaces are toric varieties. The example of how to draw $\mathbb{CP}^2$ as a triangle with tori above each point is given there. What this amounts to is remembering only the amplitudes first and then realizing that you forgot the phases and fixing that later.
So instead of drawing a line segment (a 1-cube) as you would for a single qubit, you would have to draw a $2^8-1$ simplex. Of course you can't do that, but you can project onto planes that you can draw several 2 dimensional pictures for. So this draws only certain linear combinations of probabilities for the $2^8$ basis states. Do several of these.
A lot of information is lost, because you couldn't draw the full $2^8-1$ complex dimensional ($2^{16}-2$ real dimensional) thing, but by throwing away information about phases and only taking certain linear combinations of probabilities, you get something you can visualize. Also you know what sort of structure you forgot along the way. Like when going from the point on the simplex back to the full complex projective space, you lost all the phases. You can draw those as points on the circle as usual, so you can recover the full information from those two diagrams.
If you want to say that some of the qubits are separable from others, then you get $\mathbb{CP}^{2^n-1} \times \mathbb{CP}^{2^m-1}$ where $n+m=8$. This is also toric and so also has a polytope that replaces the $2^8-1$ simplex. If you want it fully separable this would be $\mathbb{CP}^{2^1-1} \times \cdots \mathbb{CP}^{2^1-1}$ 8 times. The polytope that replaces the $2^8-1$ simplex there is a product of 8 1-cubes as already mentioned. Again above this polytope are some phases you lost along the way, but those are easy to draw in an accompanying diagram.