The British Museum is located a short-ish (20-30 minutes) walk from Big Ben, so the distance isn't a problem in seeing it. But the British Museum is surrounded relatively closely in that direction by streets with buildings of 4-5 floors, which block the view.
I can't entirely rule out that there may be some spots in the grounds from which you can just about see the top of Big Ben poking above the other buildings, but you certainly won't get a good view of it. (Arguably, you can't get a good view of it from anywhere right now, as pointed out in the comments it's currently covered in scaffolding due to renovations)
Some comments on the question suggested it may be visible from the roof of the British Museum. That seems likely, based on google maps 3d view, but I see nothing on the British Museum's website to suggest roof access is possible - and even if it were, the distance is such that I expect it would be difficult to get a good picture of you with the tower anyway.