I don’t just leave my headphones on my desk because of ants wandering around the house

They will be fine.

If you have some silica gel packages lying around, put a couple or so in there.

Is it alright if I have none or should I go out to buy some

It is alright if they are dry, but if they are not, having silica gel packs in there helps in absorbing the moisture.

Put them in a cloth bag. Sealed bag without silica not a great idea.

Alright to just store it in another bag? Not airtight or cloth but like a pouch bag type bags

... get some ant traps if they're so bad they're crawling up your desk.

Saw on my computer :(

Oh yeah, would it be alright if I were to just put it in a backpack I wasn’t using? I’m planning to get a container or case for my headphones sooner or later but would it be better to put them in a backpack or in this ziploc bag

Yeah, should be fine if it's not sealed tight

I suppose. I put one of my phones in a string bag as well, but I still put a pack of silica gel in it as a precaution.