“I can barely hold my eyes open.”

You're driving late at night, and you're sleepy. You feel like you're going to fall asleep. You tell the person who's riding with you:

I can barely hold my eyes open.

I can barely (do something)

If you "can barely" do something, it means that you are able to do it, but you almost can't do it. In the example above, the speaker is able to hold her eyes open, but she feels like they might close soon.

Here's another example of "I can barely ___" used to talk about something in the past:

I could barely keep a straight face.

hold (your) eyes open 

To "hold your eyes open" means to stay awake, and to keep your eyes from closing. This phrase is mostly used in sentences like the one above, where the speaker is saying that she can't hold her eyes open.