I'm not sure what the meaning of the question is, and I need some help understanding what I'm even trying to do. The unitary operator in question is in $U(4)$
The question is: "Decompose the above unitary operator as a product of unitary operators, each which acts only on a two-dimensional subspace of the Hilbert space"
My confusion comes from "acts only on a two-dimensional subspace of the Hilbert space" - what does this mean, and what does it imply I should be doing in my question? Does it mean I need to use matrices with columns 3 and 4 filled with zeroes?
I'm confused and my lecturer is terrible. Please help.
Fleshing out the comments in a bit of detail.
I assume $U(4)$ is the group of $4 \times 4$ unitary matrices acting on the Hilbert space $H = \mathbb{C}^4$. I like using $U$ for unitary operators, though, so I won't write $U(4)$ again (When I say "unitaries" without other qualification, I just mean elements of that-named set.)
If $U$ is the unitary in your question, in the "likely scenario" identified in Semiclassical's comment above, the question is basically asking to to find a basis in which you can write $U$ as a block sum of $2 \times 2$ unitary matrices, i.e., a unitary matrix $V$ (representing a change of basis) for which $$ U = V^* \begin{pmatrix} U_1 & 0 \\ 0 & U_2 \end{pmatrix} V, $$ where $U_1$ and $U_2$ are $2 \times 2$ unitary matrices (and $0$ denotes the $2 \times 2$ zero matrix).
Put another way, the question is to find a coordinates in which $U$ acts separately as unitary transformations of the first two coordinates and the second two coordinates.
To connect this to the phrasing of the problem, if you have a factorization of this form, then $U$ is a product of two unitaries, $W_1 = V^* \begin{pmatrix} U_1 & 0 \\ 0 & I \end{pmatrix} V$, and $W_2 = V^* \begin{pmatrix} I & 0 \\ 0 & U_2 \end{pmatrix} V$. In the coordinates of your new basis (which probably won't be the standard basis), $W_1$ leaves the second two coordinates alone, while $W_2$ leaves the first two coordinates alone.
Or in slightly more spatial terms: let $S$ be the span of the first two elements of your basis. Then the span of the second two elements of the basis is $S^{\perp}$ (see comments above). Since $W_1$ fixes $S^{\perp}$, it may make sense to say that $W_1$ "acts only on $S$," and similarly it may make sense to say that $W_2$ "acts only on" $S^{\perp}$.
It may mean that your instructor meant something less restrictive, where the two-dimensional subspaces for your unitaries may not need to be complementary to one another. But I agree with Semiclassical that this is the most likely interpretation.