If you were talking about the relatively small place I worked (15MW electricity+15MW heat extraction CHPS power plant) then it could work for quite some time.
Usually any size power station is connected to the grid, and a large amount of the economic viability is tied to the price per unit of energy that the grid needs to keep consumers supplied - the greater the demand the higher the price, generally speaking(apologies to Californians
)
The result is that power stations are often only partly run for particular parts of a day and there is usually not much need for them to run continuously for extended periods of perhaps several weeks.
This means that the fuel feed system need not be designed to operate unattended which would be costly for the kind of machinery needed to ensure fully automatic operation.
At the place I worked, we oftentimes did not run connected to the grid distribution system, and the grid price of power was nearly always significantly more than we could produce for ourselves, especially as we were able to utilise the heat byproduct which makes it much more economic compared to traditional coal fired stations which had no use for the surplus heat(in most cases).
The result was that we could run for long periods without supervision, our engines run on natural gas, with a squirt of diesel fuel required for the bang.
Our only limit would either be the amount of diesel in the tanks, which were pretty large, several hundred tons in each of them, or maybe the amount of lubricatiing oil required.
These engines use up lots of lubricating oil, and they lose it literally by the ton.
I don’t remember the rate of usage, but once a certain oil level was reached then the engines would shut down automatically. We would never actually let this happen except as part of a engine safety shutdown excercise when we would simulate low oil levels by diddling around with the oil level switch actuator arm(float switch to you!)
Such a plant could run for a week with no trouble my guess is probably for some time longer than that, most of the time we would shut down in the night when the grid price of electricity tends to fall below our break even price.