Google Sites Header Size, Banner Size, Image Max Width
The Different Types of Page Sections and Images
Header: Only one header on a Sites webpage. It can either be Banner, Large Banner or Cover Banner.
Section Image: Each content section can have a background color from your theme or upload an image.
In Page Image: Uploaded from your computer or Google Drive, these are images added into the webpage content area, within sections.
Header Image Size
Max dimensions for a header image are 2048 pixels wide or high. This means your images will render at full definition for most screens, including desktops operating at 1080p (1920x1080 pixels). If you upload an image larger than 2048 pixels, it will be automatically reduced and compressed by Google Photos, which seems to be the application handling these image uploads.
Max dimensions are 2048 pixles wide or high. Similar to the header image, this is typically a 100% browser width displayed image. Header and Section background images will be less than 100% if using a custom site width and frame option (this is found within Spacing options). Keep in mind, most desktops screen sizes are 1920x1080.
In Page Image
Max dimensions are 1600 pixels wide or high. There have been cases of users uploading very high/tall images into Google Sites and these become distorted because of the extreme dimension restriction. Best stay under 1600 pixels in both width and height at all times for full definition.
On comfortable density with default site width, the 12 grid system accomodates a max width of 1154 pixels. Therefore, if you are using these default options and aiming for pixel perfect renderings, shoot for 1154. Below is a picture of default width settings and dimensions.
Image Carousel (Slideshow/Slider)
The best dimensions for any image uploaded into the image carousel are limited to 2048 pixels wide or high. If you exceed this, the image will be sized down.