Can I run a site on my server under a user account that can't access any other sites or folders apart from its own and its children?
Yes certainly, you need to adjust the identity of the applicaiton pool that your site runs in.
Your best option would be to create an account that has the permissions you need, then create a new application pool based on an exisiting workin one. Then alter the identity of the pool to use the restricted account you created. The steps for doing this are as follows: