Forgive me if I am unclear... I haven't taken a biology class in 6 years.

I've been reading some of the responses for questions that have involved latent viral infections (chicken pox, herpes, etc).

I've also read from wikipedia that a typical cell is 10 micrometers in diameter and most studies viruses are between 10 nanometers to 300 nanometers in diameter.

It seems apparent to me that even when you account for internal structures like the mitochondria and the nucleus, a cell has a lot of internal space remaining for latent viruses to hide in, but this space is nonetheless finite.

If a person lived long enough, would their cells become so bloated with different latent viruses that their body would begin to self-destruct as their cells popped due to internal stresses?

When a virus goes into a latent phase, it is not the actual virus particle itself that hangs around. Instead, the viral DNA incorporates itself into the host cell DNA. Then, when the viral genes are activated later, the cell starts making new virus particles that are then able to infect new cells. Also, your body may be host to many different latent viruses at once, but any given cell will only have DNA for a few of them.

Also, the 'empty' space you refer too is not actually empty. That space is known as the cytosol and is crammed full of proteins and different chemical compounds.