Yes you can create your own private Blockchain on the web using BaaS (Blockchain as a Service) available -for example- on Microsoft Azure or IBM Bluemix, or Amazon AWS Marketplaces.
And yes you can store data on your Blockchain,, the data will be stored permanently on your Blockchain, you can do so on an Ethereum Blockchain using smart contracts which are a type of programs (written in Solidity -usually- and compiled to byte code) that holds data (objects) and methods that controls the data (like retrieving the data).
After the deployment of your contract, you will be able to communicate with it using it's address, you can retrieve your data, or alter it as you wish, but remember that once the contract is deployed it can never be altered or modified.