Weebly offers this type of functionality with their Paid Members app. The cost for this ranges from $4.97 – $24.97 per month.
WordPress also offers this type of plugin. Using WordPress takes a little more technical expertise, but is also less expensive and more customizable. Here is a comparison of the 5 top WordPress membership plugins. You will might even be able to find a free plugin that fits your needs.
Hope this helps!