nut milk bag from Amazon for about $12 because I couldn't find them at any local health food store. Tonight, at Publix, I found similar-looking bags selling for $1.99 for a three-pack. They are labeled as produce storage bags, but I picked them up hoping they might work for straining nut milk and juice.
Upon closer inspection (meaning . . . when I was finally able to open them), they are definitely not as fine a mesh as the nut milk bag I ordered. However, I decided to give them a try anyway.
I used them to make some cashew milk, and the milk came out very thick and frothy. Thinking it must be the larger holes letting out more pulp, I restrained it through my nut milk bag - and it came out the same way. Maybe cashew milk is just thick???
In any case, I'm definitely going to continue to experiment. Even if I use two together to strain out even more, they're still a great price, and I'm sure they'll be useful. I'll definitely report back on how they work with almond milk (since I'm more used to that consistency) and juicing.