Yes it is possible. Hep C seems content to do very little for years then the wheels fall off.
The liver enzymes we measure are actually part of the "guts" of the liver cells - when these cells die we see the liver enzymes released and can measure them in the blood.
One way the body normal deals with viruses is by killing the cells that contain the virus. In some patients they have a robust response and the disease is cured, in others the response is not enough to cure, but enough to cause ongoing damage.
Some patients, often with very high viral loads just allow the virus to do its thing and, as a result, don't have much happen for ages.
When we look at the patient demographics we can see lots of people in their 40s 50s and 60s but very few who make it past 70 - this is because Hep C tends to chop the last 10 years or so off the expected ~80 years of age average when we tend to expire of natural causes.