[Galendor]

As I read it, Middle Earth was a relatively undisturbed place with true wilderness. Much of this environemnt, such as the lands between Hobbiton and Rivendell, seems to be composed of a patchwork of forests, streams, rivers, and natural open fields. In such a landscape there would be lush and diverse wild animals and plants.

I don't think Strider in the context of the books would choose to hunt larger game requiring a bow, such as deer or bear. These animals have a lot of meat, more than can be used by a small party before it spoils. So I imagine he would primarily hunt small game, such as rabbits, squirrels, and large birds such as pheasants or grouse. I imagine such animals were far more abundant and less wary in the wilds of Middle Earth than we can imagine nowadays, and Strider as a Ranger new their ways and habits very well. So I think he could have hunted them with a simple sling (strap of leather with a pouch for holding a stone), or even caught them by hand (fat rabbit is no match for stealthy Ranger). It isn't hard to imagine he was a very skilled shot with a sling. He may have hunted such small game while traveling, in a catch-as-can fashion and taking advantage of opportunities that presented themselves. Tolkien didn't usually elaborate on such smaller concerns of daily necessities.

So I imagine small game could be "caught" in little time by Strider, carrying only a small sling in his pocket. He could also have a hook and line for catching what must have been abundant and unwary fish (bass, trout) in ponds and streams. Also it is clear in the books that Strider knew much herb-lore, and so I think he also was adept at collecting edible leaves, fruits, and roots from plants that would be good for eating.

So I don't think Strider needed or carried a bow in his general ranging duties. But feeding a small band of hungry hobbits was probably tough even for him!