He can't handle it = Δεν το αντέχει (emotionally)
= Δεν μπορεί να το χειριστεί (he's incapable of handling it, he doesn't have the skills)
I can't deal with this right now. = Δεν μπορώ να ασχοληθώ με αυτό τώρα. (no time for that now)
= Δεν μπορώ να ασχοληθώ με αυτό αυτή τη στιγμή. (no time for that now)