If you are 'short on/of something' it means that you don't have enough of something. 'Short on space' means that you don't have enough space in which to put the things you need. I'm presuming that the speaker is talking about packing a hand-luggage bag for a flight. They don't have enough room for a full-size set of headphones in their bag, so they're planning to take earbuds instead.

"Short on space" means that there is not any room for an object. Example: if you have an apartment that is very small and you had a piece of furniture that you like ( couch, chair etc), but that item will not fit in your apartment because that apartment is too small . You would say "I like that chair, but I'm short on space so I can' t buy it". This means that the chair just won't fit in your apartment because the apartment is so small. I can't think that this would ever apply to earbuds since they are so small..