Obviously, I want to get the answer without saving the page. It seems basic, but I can't figure out how to do that. If it requires a plugin, it's ok. It would be nice if the solution would also cover styling sheets, scripts and images, associated with the page.
Open Developer tools (Ctrl+Shift+I or Settings Icon at the top-right of your browser window =>
Developer tools) on the needed page, switch to the Network tab and reload page.
In the Size column you'll see the size of everything loaded (Documents, Stylesheets, Images, Scripts, ...). You can enable filter to help you find out only needed stuff at the bottom-center of Developer tools frame.
I have created my 3rd Chrome extension Page Size Inspector that quickly reports page size, cache usage, network requests and load time of a web page in a convenient way.
There's an extension called "Quick source viewer", which caches all the resources when a page is loaded. Open the viewer (blue icon), the size (number of bytes) is shown in the navigation pane on the left.