It is feasible that a geothermal power plant could be built that ran and lasted for ten millennia.
On the engineering side it would have to be an extremely robust and readily repairable piece of technology. The subterranean parts of the plant would need to be accessible to robotic repair and maintenance systems. Currently we don't have exactly that level of robotics, but that is not inconceivable that it could be developed in future.
Of course, anyone, nation or power company, wanting to build a ten millennia lifespan geothermal power plant would invest in the necessary technology to keep it working for that duration. This will be extremely difficult to achieve, but it's not impossible. Also, the repair and maintenance robotic technology can be expected to improve during its operating life.
Are there any problems with maintaining a source of geothermal energy? The answer to that is simply no.
Geothermal energy is the heat from the Earth. It's clean and
sustainable. Resources of geothermal energy range from the shallow
ground to hot water and hot rock found a few miles beneath the Earth's
surface, and down even deeper to the extremely high temperatures of
molten rock called magma.
Almost everywhere, the shallow ground or upper 10 feet of the Earth's
surface maintains a nearly constant temperature between 50° and 60°F
(10° and 16°C). Geothermal heat pumps can tap into this resource to
heat and cool buildings. A geothermal heat pump system consists of a
heat pump, an air delivery system (ductwork), and a heat exchanger-a
system of pipes buried in the shallow ground near the building. In the
winter, the heat pump removes heat from the heat exchanger and pumps
it into the indoor air delivery system. In the summer, the process is
reversed, and the heat pump moves heat from the indoor air into the
heat exchanger. The heat removed from the indoor air during the summer
can also be used to provide a free source of hot water.
If the hypothetical ten millennia geothermal power plant uses hot dry rocks as its source of heat, they are not only ubiquitous but it would guarantee a constant power supply. This technology is in its infancy, but the builders of a ten millennia geothermal energy system will have developed it because of its reliability and stability as a power source.
Hot dry rock resources occur at depths of 3 to 5 miles everywhere
beneath the Earth's surface and at lesser depths in certain areas.
Access to these resources involves injecting cold water down one well,
circulating it through hot fractured rock, and drawing off the heated
water from another well. Currently, there are no commercial
applications of this technology. Existing technology also does not yet
allow recovery of heat directly from magma, the very deep and most
powerful resource of geothermal energy.
The concept is feasible, doable though not now but certainly in future, and offers long-term stability and reliability as an energy source.