Learn English – English proverb for when a solution comes too late

For doing something that is too hard, or doing too much of it at one time, I'd say you could go with

Biting off more than he/she can chew

which usually implies "taking on too much/too may tasks", but I think it's fair to stretch it to "taking on a task that's too hard for you". This is, to me, the most obvious idiom for this, but it's fairly cliche.

You could also use

Getting in over his/her head (as in, the water is too deep)

which implies swimming somewhere where your feet can't reach the bottom, and yet you're not a strong enough swimmer to handle it. You can also use

He's out of his depth/He's in too deep

which is the same metaphor. Neither of these imply failure, since the speaker may have misjudged the swimmer/chewer. I can't think of a phrase that combines these with the notion that the task is impossible; usually impossible tasks are impossible for everyone and not just for one person (swimming in deep water may be impossible for him, in which case he's getting in over his head, but it's not impossible for me, in which case I just swim there).

Here is a related line from Shakespeare that has become idiomatic:

The lady doth protest too much, methinks.

It comes from Hamlet and you can read all about it here. As the Wikipedia article suggests, it has come to mean that one can "insist so passionately about something not being true that people suspect the opposite of what one is saying."

I have heard it in many forms, such as "methinks the lady doth protest too much," "the lady protesteth too much," or more simply "she/he protests too much."