To have it hidden from search engines - add a robots.txt with the following content:
Place the robots.txt in the root directory of your website/application.
To make it available for others to see at a competitive rate, put your site on Heroku or perhaps create a droplet on Digital Ocean for £5 a month. You do not need a domain for either.
- Heroku you name your app, and you get a Heroku subdomain that contains the name of your app (if I remember it correctly)
- Digital Ocean you can share the IP of your droplet and make sure you have default configuration file set if on Nginx - not sure about Apache as I do not use). Let me tell you that it looks more professional however if you use a domain (perhaps a subdomain that makes sense).
Alternatives to Heroku and Digital Ocean is Google App Engine (slightly more expensive), or any other Web Hosting Provider you can think of (I don't like the dependency on them - that why I prefer the big 3s).
On LinkedIn you have a section titled "Accomplishments" -there you can list it under "Projects" perhaps? Unless you created your project at Uni, School or Work - then you may wish to share the link under the appropriate section.
The source I would share on GitHub or BitBucket - creating a public repository everyone has access too (which makes sense to supply if you wish to impress with your programming skill).
Most importantly highlight both - URL and Repository on your CV :)
Generally, Employers looking for Developers like to see your GitHub account to see how active you are and filter your contributions towards the community - it tells a lot about you as a developer. ;)