The distance aside, the terrain is extremely difficult, and in most places, impassable without very specialized mountain climbing gear. Water is plentiful, but food is scarce. There are elk and deer in the lower elevations, but there’s a reason that even the natives didn’t live in there.
In the late 1800s, most of the interior of the Olympic Range was still unexplored, and the Seattle Press sponsored a team to explore the area (they went from north to south). Called the “Press Party” the group, although well equipped, barely made it out alive. It is believed that some of their equipment caches are still out there. If you can find a book on the expedition get it, it’s an incredible story of real survival.
James Christie's party of six men (five completed the trek), four dogs (including Christie's two bear dogs, Bud and Tweed - the other dogs were Daisy and Dike), two mules (Jennie and Dollie) and 1,600 pounds of supplies spent five-and-a-half months in the mountains on a north/south crossing December 8, 1889.
For those of us that hike the area, including going off trail, the Olympics are no place to mess around. This is real terrain with real dangers. Trying to walk out without a plan other than ‘go west’ will kill you. Get lost in the forests, especially the old growth forests, and finding your body will be like looking for a needle in a haystack.
For comparison, the Olympic National Park alone is roughly the size of Rhode Island! That does NOT include the Olympic National Forest lands and a wilderness area or two thrown in. Your direction of travel will depend entirely on where in the Olympics that you're lost.