According to Star Trek technical manuals I've found, and other resources from various places, the Sick Bay systems are completely separate physically from the rest of the ship. It''s got its own power supply, it's got its own replicators, it's even got it's own Emergency Power. It is for all intents and purposes a complete separate system. The only reason the Doctor is at all aware of anything that goes on on the ship is because in Episode 01x02, Janeway has the Doctor's program linked to all logs for everything that happens ship wide. That way he can be made aware of any emergencies that could come up. So therefore the separate system is networked with the rest of the ship.

Also, Sick Bay's systems are mainly only capable of storing one maybe two EMH's at a time. Any other holograms are taken from the main computer's memory, or the holodeck's memory storage. Assuming the doctor needed a secondary back up, he could instruct the computer to transfer a copy of his program into the holodeck, and use It as a secondary Med bay. The only problem with that is that in emergency situations upon all starships, the first systems to be deactivated to reroute power is the Holodeck. Which is done by the computer. This can be seen in TNG Episode 3x06 when the power on the ship is waning and the computer shuts down the holodeck because of it was Standard Operating Procedure.

It must also be taken into consideration that the EMH by his own admission has the sum of almost all the Medical Knowlege known to Starfleet. In order to be able to serve a star ship in the event that something like him would be necessary, and considering the types of encounters most exploration ships have, this would be absolutely necessary.

The size of all of this information would be tremendous, not to mention the computer in Sick Bay most likely draws upon His programmed knowledge when asked a question by the standard medical officer who would normally be there. It would be a waste of space to have both his knowledge, and a copy of this knowledge in the system at the same time.

And since Sick Bay is as stated before a separate system designed to be able to sustain itself in case of a ship wide power failure, or computer failure, we must assume that the amount of space within the computer must be finite.