Is it possible to fly horizontally with zero angle of attack?
Unless you tell us which book you read this in (and what exactly it said), it is not possible to answer your question (the answer depends on whether the zero angle of attack is that of the airframe or the airfoil (wing)). We can try for a general answer though.

As far as angle of attack is concerned, there are two kinds- geometric and absolute. First, angle of attack is the angle between some reference line and the freestream (i.e. the velocity vector). The selection of reference line determines the 'kind' of angle of attack. 

Geometric angle of attack, $\alpha$ is the angle between the chord and freestream; this is the one usually used. Absolute angle of attack, $\alpha_a$ is the angle between the freestream and the zero lift line (the line, which when parallel to the freestream, produces no lift in the wing).

The geometric and absolute angles of attack can be related through $\alpha_a = \alpha - \alpha_{L=0}$, where $\alpha_{L=0}$ is the angle between the chord and zero lift line.



Relation between geometric and absolute angles of attack, image from gtae6343.wikia.com

For a cambered airfoil, the zero lift line is negative i.e. the airfoil will produce lift even when the geometric angle of attack is zero (in this condition, the absolute angle of attack is positive). For a symmetric airfoil, of course, the zero lift line coincides with the chord and as such, the airfoil produces no lift when the geometric (or absolute) angle of attack is zero.

As far as the lift is concerned, what matters is the absolute angle of attack, while we usually use the geometric one. Simply put, what matters is the absolute angle of attack of the wing. If this value is positive, lift will be produced and level flight is possible, whatever the aircraft attitude.