There are physically separate networks for classified and unclassified e-mail. There are also separate cable and pouch channels.
Source: I worked there.
State was not very proactive in IT focusing well into the 1990s on centralized work processing and administrative applications run on WANG VS terminals. It was a sign of status among senior staff well into the 2000s to not use a keyboard "like a secretary". The people managing IT were a bunch of turf conscious bureaucrats who had started as teletype operators.
It wasn't until Colin Powell lit a fire under the ass off the IT Bureaucrats that rank and file employees got access to basics like personal e-mail accounts and Internet browsers, but everything went through a poorly run firewall that would kill any incoming mail with a .pdf or other attachment.
It wasn't like an ISP you could dial into from home, you need VPN software and justification from a department head to get it and there was a very expensive internal reimbursement to it IT Lords to pay. Upper level management got department issued official Blackberries which were linked to the State e-mail through a firewall.
Relatively few have access to the Classified e-mail; it's on a need to know basis and there are a lot more restrictions regarding were and how it can be used.
Clinton travelled more than any Sec. State. Had she been forced to rely on the State classified or unclassified e-mail systems alone she couldn't have done her job effectively. I have no way of knowing but would guess she had access an officially issued Blackberry when traveling domestically but chose not to use it, using one set-up to her private domain instead.
I was in the Foreign Service starting in 1992 with a background in IT but working managing printing at USIA. IT was so backwards there in '82 I went down the street and spent $1300 of my own money for a Radio Shack Model 100 portable I carried to work with me. In 1987 I setting up physically a separate Mac centric LAN for publication production and administrative applications in our DC office and overseas in 1990s at our production facility, connecting it to the Internet with a separate .gov domain in 1994. I ran all official communications for our operation in and out though databases via a single central e-mail address so it was archived better than the official e-mail, which at the time was still only available on WANG VS mini computer workstations.
This is all a tempest in a teapot for political reasons which overlooks the fact that when anyone important has anything to say off the record they don't type and leave any record they pick of the phone and talk to the person. But then the NSA has a record of that.