This is a multi-step process that will involve changes in the theme and on the product level.
Step 1: Theme changes.
From the left navbar, navigate to Theme> Edit Theme > Shopping Cart > Basic Settings
Now, uncheck the box labeled Show Shipping Calculation.
This will hide the shipping options on the cart page:
Step 2: Set Product to ‘Does not Ship’
Navigate to Catalog > Products > Edit Product > Shipping
and check This Item Is not Shipped:
(You can make these changes en mass via a product export/import. Export the four columns ItemID, ItemNumber, ItemName, NonShippingItem. Open edit the exported file and set the value the column’ NonShippingItem’ to true, save the file and reimport it.