I can't quite tell what you are trying to achieve here, or why the form needs clearfix at all. If I remove it, everything seems to work fine. Can you give some kind of idea of what you want the layout to look like?
If you type anything but an email address into the form it will generate an error message under the form, which 'sticks out' of the form container. I want this to be contained within the FORM, or an enclosing DIV or so.
(Remove the 'clearfix' class and, type something into the field, other than an email address, and then leave it). It's then obvious that it sticks out and interferes with the content below.)
However, if I try to make the enclosing FORM (or a div) as tall as the floated contents it takes the height of the ASIDE element...
Overflow: hidden is a common trick for forcing a container to wrap around its floated content. I can't quite remember now what was happening before, but I think it was the "clear: both" on the clearfix class that was getting in the way in this instance. Normally if one float containing method causes a problem, I just use another. There are lots of pages online that detail the various containing methods, but here's one I did a while back: