There is an (apparent) event horizon in any accelerating frame: that is, if you are accelerating and keep accelerating then light from some distant events behind you will never reach you. The distance from the spaceship to this horizon is $c^2/a$, where $a$ is the proper acceleration of the spaceship; see Wikipedia for details (look under "apparent horizon of an accelerated particle"). Of course this horizon only persists as long as the spaceship keeps accelerating, which in practice isn't very long compared to the distances involved, so we never notice it in everyday life.
Note that this apparent horizon differs quite substantially from the event horizon of a black hole, in that it is only a horizon for the observer in the spaceship. This is to be expected. The equivalence principle has a lot of caveats that are very important in practice: a uniformly accelerating reference frame is equivalent to a uniform gravitational field in a small enough region of spacetime. Acceleration is thus not "the same" as gravity, but it acts "like" gravity if you only consider a small enough region of space and small period of time. Real gravitational fields (such as produced by a black hole) are very far indeed from being uniform.
