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. The existence of such an operator between the two above-mentioned spaces was first established by Pelczynski. 

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, who proved the existence of the isomorphism. Anyway, I think that this also qualifies as an example.