How can I hide the order status from a site user on the order history page of the site?

A site user will always be able to see the current workflow status of an item when they check the details of the order in their order history on the site.  However, you can modify what they see by setting up a custom workflow for a product.  With a custom workflow, you can leave the status the user sees blank which will in effect hide the status from the user. 
To set up a custom workflow, you will need to go to Preferences > Workflows and choose "Create New Workflow".  You will need to first give the new workflow a name and add any applicable instructions before clicking "Save".  Then you will need to create a stage for each valid step in the workflow.  For each step, you will need to define what the "Internal Stage Name" is for each step.  The Internal Stage Name will be what you will see as an admin when you are processing orders.  If you would like the customer to see an alternate name for that stage, you can modify the "External Stage Name" - what the site user will see in their order history.  You can also choose to leave the External Stage Name blank and it will show as a blank entry in the site user's order history.
Once you have created all of these steps, you will need to define that the product will use the new workflow by editing the product in Sites > [Site Name] > Products > [Product Name] > General tab in the "Order Management Section" and choosing the workflow from the setting "Order Status Workflow".  Once set, orders that come in for this product will show no status on the site to site users when the product is in a status with a blank External Stage name.
Note - custom workflows is an advanced feature not available with the Pressero Standard plan.
For detailed information on Custom workflows, please refer to this article: Ch. 022. Workflows (Custom Status options)