“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 this to the person riding with you.
I can barely hold my eyes open.
If you "can barely" do something, it means that you are able to do it, but you almost can't do it.
For example: if you need to move a heavy piece of furniture, but it's hard for you to pick up, you can say:
There's no way I can move that! I can barely lift it.
Say that you "could barely ___" for continuous actions and situations that you had trouble with in the past:
When I first moved to Mexico, I could barely speak Spanish.
To "hold your eyes open" means to stay awake, and to keep your eyes from closing:
Look at Jamie. He can't even hold his eyes open. Put him to bed.
This phrase is mostly used in negative sentences like the one above, where the speaker is saying that she can't hold her eyes open.
You could also say "keep your eyes open":
I can barely keep my eyes open.