As of Rust 1.38 (released in September 2019), a cleaner alternative to previously posted answers is possible using an intermediate const initializer. This approach works for arrays of any size:
const SIZE: usize = 100;
const INIT: Option<Box<Thing>> = None;
let array: [Option<Box<Thing>>; SIZE] = [INIT; SIZE];
(It works with or without the Box; the example uses Box because it was used in the question.)
One limitation is that the array item must have a default representation that can be evaluated at compile time – a constant, enum variant, or a primitive container composed of those. None or a tuple of numbers will work, but a non-empty Vec or String won’t.