The majority of pets you see would never have rabies, even if they were unvaccinated, but it's still good to have a healthy sense of worry over it. You can receive the rabies series of 3 vaccines for yourself (although it's expensive and it may be difficult to find a place that will do it. It's not a common vaccine for people to get a pre-exposure rabies series). You would still need to receive treatment if you got bit by a potentially rabid animal, however, but it would be less vaccines that you would need at that point. Veterinarians are vaccinated against rabies at least once in their lives.
However, that's unlikely to actually help the issue with hypochondria. Hypochondria doesn't always listen to reason. This is something that you may want to address with someone in your life who you can trust.
It also helps to make a plan, so that you know what to expect and what's going to happen. Talk to your vets/staff about it and what they would do if a dog/cat did seriously bite someone and the pet was not vaccinated against rabies. Chances are high that you would have to fill out some paperwork and would go straight to the hospital. The medical professionals there can be trusted to make the right decisions regarding whether you need post-exposure rabies or antibiotics, etc. There are laws regarding rabies and they depend on what state you live in. You can look up those laws on the veterinary state board website for the state that you live in. This will tell you how the animal needs to be quarantined, vaccinated, etc. after the bite.
Rabies treatment is typically not undergone if you just get a scratch. Rabies is spread by saliva. Technically saliva could be on the nails of the animal when it scratches you, but that's stretching it a little bit. Rabies typically spreads by a bite. Also, always wash a bite or scratch with water and soap really well after it happens. This can actually reduce your chances of getting rabies because the virus can be washed out of your skin so always do that before you do anything else!