In Google sites you can Insert your own specific Google Calendar.
"Insert" > "Calendar" > Select your specific Calendar
Embed into any html page
The Calendar selection popup is common on most forms these days (Hotel booking, Airline etc..)
...Yet Calendars rarely display LIVE information directly on the selection calendar (Sold out flight, Fully Booked Day etc...)
Here is a "Guest House" booking demonstration of a public Google Calendar with Date Information on the calendar selection.
This is achieved by combining the following Google elements:
Hidden 'Booked' Date Selection
Google Gadget Source Code XML (Open and View Source)
Google Gadget - Main Goal :
Live Calendar > Form > Show Price Details > Online Payment Option (with selected details)
Here is a more advanced version with a Date Calculation (From the Live form submission).
You can validate forms within Google Docs even with a manually created form.