Considering that dying "from old age" actually means that one dies from an illness related with aging, is it inevitable to eventually acquire diseases in old age?
There is a thought that aging itself should be considered a disease. In which case, death by disease is inevitable with our current level of medical technology.
During the last 25 years, by targeting the underlying processes of aging biomedical scientists have been able to improve the health and lifespan of model organisms, from worms and flies, to rodents and fish. We can now consistently improve the lifespan of C. elegans by more than ten-fold (Ayyadevara et al., 2008), more than double the lifespan of flies and mice (Bartke et al., 2001; Sun et al., 2002), and improve the lifespan of rats and killifish by 30 and 59%, respectively (Valenzano et al., 2006; Zha et al., 2008) (see Figure Figure11 and Supplementary Table 1). Currently, our treatment options for the underlying processes of aging in humans are limited. However, with current progress in the development of geroprotective drugs, regenerative medicine, and precision medicine interventions, we will soon have the potential to slow down aging (Bulterijs, 2011, 2012).
Aging is not synonymous with disease, and hence it is generalized as such, although vulnerability to diseases does increases over time.
Why is it not a disease? Well biological aging can be defined as:
A process of progressive, intrinsic, and generalized physical deterioration that occurs over time (begins at the age of maturity).
While a disease is:
An abnormal condition of an organism which interrupts the normal bodily functions that often leads to a feeling of pain and weakness, and usually associated with symptoms and signs.
Now let's look as to why we cannot live forever and as to why as we get old our disease susceptibility will be (in general) increased, meaning that yes the majority of the population will be dying from a form of a disease. If not a disease it will be a failure of the body system to maintain life.
I will be looking at this in the form of Natural Selection.
The force of Natural Selection declines with age. For example, a gene that is beneficial during young age, but bad during old age (such as making you susceptible to several diseases) will spread in the population because it will manifest after you've reproduced.
Example#1: A gene that helps achieve good blood coagulation and prevents hemorrhage in young people will spread in the population. However, in old aged people, it will cause an increased risk of stroke. Such associations have been previously found for many genes associated with blood characteristics.
Example#2: Overproduction of sex hormones in young age is beneficial for reproduction but will cause prostate and ovarian cancer later in life.
This is called Antagonistic Pleiotropy
In addition, Natural selection is not concerned with late acting mutations.
So as we age will we be afflicted by a disease? Most likely yes, it depends on the trends in disease prevalence which have changed throughout the years due to preventative medicine. If we do not acquire any death-inflicting (be it slow or fast acting) disease, our body will give up before we do get one.
If you'd like to look at disease prevalence, chronic illnesses throughout the years, or the specific trends in a year, just go over to the CDC's webpage.
"Old age" diseases are fundamentally malfunctions because stuff breaks. Some blame this on shortening telomeres, although an equally big part of the problem is that there are way too many important pieces that are unable to regenerate themselves.
Think of it as a machine where you can't normally get replacement parts. Stuff wears out, goes off warranty, and breaks.
Experimental regenerative medicine is working on fixing that with 3 fundamental approaches:
Adult pluripotent stem cells/regrowth. Researchers are looking for growth/regrowth/cell proliferation and differentiation mechanisms that already exist so they can be leveraged to regrow stuff that normally doesn't regrow. The advantage is that all the parts are grown from your own DNA or cells which means minimal problems with rejection or incompatibility. As an added bonus it could be considered "all natural". The main risk is cancer due to misconfigured cell growth/proliferation.
Lab-grown or 3D-printed replacement. Higher risk of rejection and other problems but simple and cheap compared to the others. Some research includes growing human-cell transplants in animals which raises various ethical and safety concerns.
Cybernetics or machine implants. Probably the most expensive and works out the best... when it works properly. "Robot" parts keep getting better and better, and have the bonus that they are more readily reprogrammable than DNA and biochemistry. A prominent research area includes brain/nerve interfaces with machine parts - expect pieces that either adapt to neural outputs or connect to nerves so you can "learn" how to use them. The main drawbacks of this approach other than the cost include vulnerability to EMPs (a massive solar flare or EMP attack will fry all unprotected cybernetics/implants/robot parts), being dependent on both electricity and regular food, risk of getting hacked, and general lack of self-repairability short of replacing parts that break (again). Alternately you could argue that the risk of getting hacked through cybernetic implants and robot parts is a fair tradeoff considering they're immune to organic viruses and infections (except if someone invents bacteria that eat through solid metal or plastic pieces)