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.