Your eyeballs are your whole eyes, rather than just the part which can be seen between your eyelids.

If you eyeball someone or something, you stare at them.

If you are eyeball to eyeball with someone, you are in their presence and involved in a meeting, dispute, or contest with them. You can also talk about having an eyeball to eyeball meeting or confrontation.

You use up to the eyeballs to emphasize that someone is in an undesirable state to a very great degree.