The Internet is mostly as you would expect from the modern day. A wild collection of files scattered across servers and computers throughout the world.
The problem is - there is no readily accessible FTL communication. It's easily minutes, hours, or months to communicate from one planet to the edge of a system - radio waves cap out at the speed of light (as they are light) - much less communicating with a ship in another solar system.
While on a mission, a ship is generally out of touch carrying it's on intra-Net with it. When the ship arrives back home it can hook back into the rest of the internet, optionally taking on uploads of new information and new files.
Fleet members may speak with the I.T. department of Engineering regarding what data stores are flagged as worth copying and bringing with on the next voyage. Bribery with caffeine is recommended.
Reports and updated files may be routed to a ship at a remote location, if a communication probe (generally based on quantum entanglement) or other extended network has been configured. Unfortunately, there's a time delay - emails work, but immediate conversations such as instant messaging or voice calls experience too many delays to be tolerable.