In general, we might ask when we can find interesting spaces $X, Y$ such that $X$ is homeomorphic to $[X, Y]$. By the Lawvere fixed point theorem $Y$ must have the fixed point property. Happily, $Y = [0, 1]$ does in fact have the fixed point property (e.g. by the intermediate value theorem), so this is not an obstruction.

I know that computer scientists have constructed spaces $X$ such that $X$ is homeomorphic to $X^X$ in domain theory, but I don't know enough about it to tell whether the techniques are relevant here.