Can I install LAMP server on Ubuntu One private cloud?

Is this possible to install apache server, MySQL database, and PHP on the Ubuntu One private cloud?

Can I use a Ubuntu One private cloud as my personal web server? If yes, then how?

No. UbuntuOne is a could personal files storage system, it will not run the services for you.

Having said that you can have a service running in your computer and have those files be stored in the cloud, that would mean that your computer is responsible for the reply to any request done in a running server (HTTP server, SQL server or anything else) and would use your stored files in the UbuntuOne.

That would have serious degradation in the performance making it hard to justify since any change done in your local files would be then synchronized to the UbuntuOne servers, 1 request once in a while is fine, turn that to a few dozen and it would just burst.

A better option for that type of features is really use something like Amazon EC2 instance and have your services running there. Ubuntu can deploy and configure that on the fly using Juju Charms.

Form more information on cloud services using Ubuntu please have a look at http://cloud.ubuntu.com/ and https://jujucharms.com/ for the charms deploy process.