Maybe each starbase could have those servers as they are widely spread enough to give coverage but should also always be within range of a subspace beacon
One problem with a Federation wide "internet" is the speed of subspace radiation.
As I remember there were about half a dozen episodes of TOS where Kirk subspace conversations with Starfleet Admirals, requiring speeds much greater than one light year per second, or 31,557,600 times hte speed of light.
And in ENT Captain Archer sometimes had subspace conversations with Earth, requiring such speeds.
But in TOS there is also "Balance of Terror" where it took Kirk hours to get an answer from the nearest command base. And in "The Enterprise Incident":
KIRK: You understand that Starfleet Command has been advised of the situation?
TAL [on viewscreen]: The subspace message will take three weeks to reach Starfleet. The decision is yours, Captain. One hour.
Unfortunately, "this galaxy ain't big enough" for real time converstions between even the closest stars (at least in the parts of the galaxy around Earth where most epsiodes areset) on one hand and a subspace message taking 3 weeks to reach some place in the galaxy on the other hand. At least the galactic disc isn't and if some episodes happen beyong the galactic disc in the galactic halo the edge of the galaxy in "Where No Man Has Gone Before" would be very far from Earth and very high warp speeds would be needed to reach it, inconsistent with "By Any Other Name".
The structure of the galaxy is rather granular, so the entire diameter of the galactic disc is only about 20,000 times the average separation between a star and its nearest neighbors in our region of the galaxy.
Of course I can think of two separate theories to explain such differences in subspace communication speeds, but they wouldn't allow for instant computer access very far beyond Federation or known space.