Another family of infinitely many examples: take $Y$ to be a reflexive Banach space which is not a Hilbert space, then $Y\oplus Y^*$ is isometrically isomorphic to its dual, without being a Hilbert space.
Theorem: Suppose that $X$ is a Banach space and $\phi:X\to X^*$ is an antilinear isomorphism. If, for all $x\in X$, $\phi(x)(x)$ x$is orthogonal to$x$(in Birkhoff-James' sense, ) to$ker(\phi(x))$, then$X$is a Hilbert space. 2 added 276 characters in body Another family of infinitely many examples: take$Y$to be a reflexive Banach space which is not a Hilbert space, then$Y\oplus Y^*$is isometrically isomorphic to its dual, without being a Hilbert space. If the isomorphism verifies additional properties, then the result is true. Namely, if a Banach space is isometric to its dual, under certain conditions, it is a Hilbert space. See Theorems 2 and 4 in http://arxiv.org/pdf/0907.1813.pdf and reference therein for similar approachesresults. An example of this kind of results is the following: Theorem: Suppose that$X$is a Banach space and$\phi:X\to X^*$is an antilinear isomorphism. If, for all$x\in X$,$\phi(x)(x)$is orthogonal to$x$in Birkhoff-James' sense, then$X$is a Hilbert space. 1 Another family of infinitely many examples: take$Y$to be a reflexive Banach space which is not a Hilbert space, then$Y\oplus Y^*\$ is isomorphic to its dual, without being a Hilbert space.