The box and its contents - creatures included - probably vanish from sight.
The rules on this appear to be... missing.
The author of the Invisibility spell description, it seems, had an oversight. At the beginning, they write:
The creature or object touched becomes invisible, vanishing from sight, even from darkvision.
... and then they proceed for the remaining paragraphs to describe the nuanced details of the workings of an invisible creature, ostensibly entirely forgetting to provide any similar level of detail on the topic of invisible objects - or, in fact, any information at all.
At the end they mention that objects can be made permanently invisible, but in addition to the fact that objects which cast light still do so despite being invisible themselves, that's about all we know on this topic.
The rules of D&D 3.5e are not exactly a model example of a complete and thorough rule set, so this is not totally surprising.
So what probably happens?
We can only really extrapolate from what happens to creatures: when they become invisible, any gear they're holding onto or wearing vanishes, and so does anything contained inside that stuff. Things contained inside the creature disappear, too, so we don't see the hovering contents of a digestive tract drifting through the air.
The closest parallel for an object would be this, I think: the object vanishes, and so does everything contained inside it. Anything that could be considered a part of that object, or an extension of it, vanishes too - the shade on a lamp vanishes just as the helmet on your head would, and coats hanging from a coat stand might vanish just like the key dangling from your finger would. Anything contained within those things - like a wallet inside one of those coat pockets - also vanishes.
So if you hid in an invisible box, you'd vanish from sight as well by the nature of being inside it.
What if you leave the lid off?
If someone glances in such a way they happen to peek straight inside the box, then... I'm not sure. I'd imagine the parts of you not hidden behind the (presently invisible) walls of the box would show up, but that's just me. That might, by extension, mean that if you're standing in an invisible vase that comes up to just beneath your chin, people would just see your apparently disembodied head floating in midair.
The rules don't go into that detail, though. Perhaps they might've if it weren't for the apparent oversight.