In several episodes, they worry about losing the Doctor. Either his program may be corrupted, or in one episode they wanted to transmit him to Earth through the long-range communication but worried about losing him. In episode Living Witness, we learn the Doctor is backed up. Why don't they back him up and restore him if things go wrong?
They don't worry about losing the medical expertise of the original Mark I Emergency Medical Hologram. The holomatrix for that entity is always safely ensconced in the computers in the sickbay, in its limited and unenlightened form.
What they worry about losing is the added shell of information that has become the Doctor for the crew of Voyager.
This is the added shell information which takes the data from the medical computers and merges it with the practice, consideration, additional programming, grafts, upgrades and real-time experience that has become part of the Doctor's holomatrix program.
This personality matrix, now a distinct and unique individual is what they have come to care about and fear losing, just as much as any organic sentient in their crew. That personality is unique and as far as I am able to tell is no longer backed up within the ship's computer.
There is a device called the EMH Backup Module which functions as a site to store backups of the EMH holomatrix.
The USS Voyager's backup module was stolen during an attack and recovered centuries later by the Kyrians.
This may also imply without said device, Voyager was no longer able to back up the EMH after the fifth year of their return to the Alpha Quadrant.
"Please state the nature of the medical emergency."
The EMH, Emergency Medical Hologram is a distinctive piece of software with access to a number of large databases of medical treatments for thousands of conditions including injuries, disease management and psychological maladies.
It is meant as a support tool to a working and living doctor whose judgements would take priority over the medical algorithms used to diagnose conditions when the EMH is in use.
The EMH Mark I, on which The Doctor's program is based, was listed as "Emergency Medical Holographic Program AK-1 Diagnostic and Surgical Subroutine Omega 323" in Voyager's memory. It was developed by a team of engineers to be an emergency supplement to the medical team on starships. Only meant to run for a maximum of 1,500 hours, it had little personality and interpersonal skills.
The EMH is a guided artificial intelligence/algorithmic diagnostic tool. It's operation the sum of the medical databases used in the Federation and her allies combined with the finest near AI programming in the Federation.
It attempts to determine the same way a doctor would by assessing a condition and using pattern recognition in an attempt to find a corresponding malady and treatment protocol to the malady based on information in its "experience" stored as data.
"I'm a program, not a doctor. Please consult an actual physician."
In its basic form, the EMH, is designed to support and assist. It was not designed to operate as a long term replacement for a doctor.
But this is what Voyager ended up needing, a long-term doctor who could provide medical support AND tracking for staff over a duration. In short, the EMH needed to become a DOCTOR, not a piece of SUPPORT MEDICAL TECHNOLOGY.
A doctor remembers. A doctor listens. A doctor is involved. Those interactions don't just become patient records, they become a RELATIONSHIP.
And those interactions are stored in the EMH/Doctor's holomatrix as permanent memories, such interactions with the crew over time, over space, would eventually become the difference between life and death as Voyager's doctor slowly began to evolve and listen and learn and connect with that crew, improving his performance over time.
So much so with the acquisition of the portable or mobile emitter, he became able to move among them, go on away missions and participate in shipboard activities, in ways far beyond his design parameters.
But it's just a program, right?
On the first day or even over the course of its first year, it was just a program, when it could not change any aspects of its interactions, it could not even shut itself down unless someone told it to "End EMH program" before leaving sickbay. It lacked volition. It lacked the power of choice.
When Kes began to work with him and Janeway began to give the EMH the power of choice, the power to make decisions, the power to override human concerns based on its medical knowledge, it began a slow trek toward a form of sentience. It is this sentience that makes the Doctor/EMH unique and potentially more difficult to be backed up.
The Voyager crew was always able to re-activate from the backup archive, the original EMH program without its unique personality, without any memory of the crew, or any of their relationships. But this being would not be the Doctor. It would be just the EMH.
This problem was challenged several times when the Doctor's acquisition of knowledge began to cause problems for the computer he was housed in, risking all of his memories due to overcrowding and lack of data storage capacity.
In 2373, The Doctor reached the limit of his memory capacity and started suffering massive memory loss. The Doctor's core programing which had become severely fragmented because of constant use of the EMH over the course of the past two years. The maximum operation for the EMH was intended for around 1,500 hours (roughly two months). To further complicate matters, The Doctor's expansion of his original programming to include personality subroutines and interests in things such as opera, interpersonal relationships with the crew and engineering skills, had filled all of his available memory buffers, leaving little room for his program to operate in its intended function.
A fusion of the diagnostic program and the original core EMH software created a new being whose capacities would be greater than the original and was UNIQUE. Further modifications of his program expanded his experiences further creating a more diverse and expanded psychological profile.
With the creation and addition of the Emergency Command Hologram subroutines, the Doctor further distinguished himself as a unique and capable individual capable of adaptation and development which would later prove instrumental in his fight for the rights of photonic AI sentients.
There is no in-universe explanation so the real reason is obviously that SciFi writers don't know the difference between software and hardware (or more likely are aware of the fact that viewers don't).
One in-universe explanation I use to suspend disbelief is that Digital Restriction Management (DRM) is widely used in the Star Trek universe and that hence the Doctor (and other holographic programs) will only run on whichever system is currently authorised for it.
They worry about losing The Doctor as I like to call him because over the past years of their 70 year trip home; The Doctor has become so much more than an EMH. he's become an ECH and he's just added so much to his program he feels like a real person inside and out. The crew feels the same way. For example Tom Paris exchanges jokes with him about his bedside manner, to which the doctor jokes back about his bedside manner. They're afraid of losing The Doctor because it would be like losing a crewmember, which has happened and The Doctor finds out in the episdoe 'Latent Image' and in that episode The Doctor gets angry ANGRY that's not something you'd get out of a hologram. My point is he isn't just some worthless thing that treats you and walks away to the next patient. He's a Doctor, a practically human doctor!
For much the same reason they don't store copies of people in the transporter buffer...in the Star Trek universe, people have energy-based souls, and you can't deliberately copy that with any technology. The Doctor has a soul. It can be manipulated a little, it can be moved around, but it can't be copied. Any technobabble answer is going to be in service to that fundamental premise of the Star Trek universe.