Yes and yes - but with certain limitations.
Firstly, we are not a hosting company and would probably not be able to offer you a competitive deal for this particular type of arrangement. If you want a basic, small volume html site (a few MiB), then there are hundreds of low-cost providers who can offer you a much better deal than us. If you want to run a much bigger site (perhaps several GiB) or you want to use MySQL and/or PHP and enjoy a high degree of control then you might wish to consider one of Bytemark's excellent "virtual machines". These behave as if they are dedicated hosts. It is a steep learning curve because you need to grasp the concepts behind running a Linux server remotely. The advantage is that you enjoy a high degree of control (and responsibility) for your site. And it's a fantastic way to learn about Linux! 🙂
Nevertheless, we will still host plain HTML/XHTML customer-created sites. Subject to certain checks, we give the customer SFTP (secure file transfer protocol) access to a so-called "CHROOTed" directory on the server. This effectively isolates the customer's site from the rest of the server. However this arrangement explicitly prohibits the customer from running any server-side scripts such as PHP.