The first example that comes to my mind is that the Banach spaces $\ell_\infty$ and $L_\infty [0, 1]$ are isomorphic (that is, there exists a linear homeomorphism of one onto the other), and yet it seems that one can't just write down an operator that provides the linear homeomorphism.

A similar example is that if $K$ is an uncountable compact metric space, then the Banach space $C(K)$ of continuous scalar-valued functions on $K$ (equipped with the supremum norm) is isomorphic to $C[0,1]$, the space of continuous scalar-valued function on the compact interval $[0, 1]\subseteq \mathbb{R}$. Thus, for example, if $\Delta$ denotes the Cantor set, then $C(\Delta)$ and $C[0,1]$ are isomorphic as Banach spaces. The proof of this relies on a result called Miljutin's Lemma. Anyway, I think that this also qualifies as an example.