> Does there exist a smooth proper morphism $E \to \operatorname{Spec} \mathbb Z$ whose fibers are Enriques surfaces?

By a theorem of, independently, [Fontaine][1] and [Abrashkin][2], combined with the Enriques-Kodaira classification of complex surfaces, the only possible surface of Kodaira dimension $0$ or $1$ that can appear this way is an Enriques surface.

In particular, no K3 surfaces are smooth over $\mathbb Z$. Hence the K3 double cover of $E$ is not smooth. The cover is etale away from $\mathbb F_2$, hence must be singular over $\mathbb F_2$. This means $E$ most be a classical Enriques surface (a $\mu_2$-torsor rather than a $\mathbb Z/2$-torsor or $\alpha_2$-torsor).

It is possible to get some other information about $E$:

The Picard group of this surface has rank $10$. The Galois action on the Picard group must be unramified at each prime, hence trivial, so the full lattice of cycles is defined over $\mathbb Q$. Thus by the Lefschetz trace formula, $E$ has exactly $25$ $\mathbb F_2$-points.

Some other questions that might be helpful to solve this one are:

> How many K3 surfaces are there with good reduction away from $2$ (and Picard rank at least $10$, and a fixed-point-free involution, etc.)?

> Given an Enriques surface over $\mathbb Q_2$, what are obstructions to good reduction over $\mathbb Z_2$, other than ramification of the cohomology?

> Which classical Enriques surfaces over $\mathbb F_2$ are liftable to $\mathbb Z_2$? Can something be said about the singularities and Galois representations of their $K3$ double covers?

> Can we compute the discriminants of [explicit families][3] of Enriques surfaces and try to solve the Diophantine equation $\Delta=1$? 


  [1]: http://www.ams.org/mathscinet-getitem?mr=1274493
  [2]: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1152&option_lang=eng
  [3]: http://en.wikipedia.org/wiki/Enriques_surface#Examples