There are two kinds of starvation: one is serum starvation, where your cells will slowly arrest their proliferation (block in G1) for absence of growth factors provided with FBS, they will remain vital for some time but, depending on the cell line, they will eventually start to die. You should be able to obtain a good starvation in 24-48 hrs with FBS free medium. Another kind of starvation is fasting, so glucose-starvation. That is more complex to carry on over time, since your cells will not have any energetic substrate to use, will suspend their metabolism, and start to die quicker as glucose reserves run out (they could also undergo autophagy to survive and take time, but not for long). Anyway, cell cycle arrest in G1, observable with flow cytometry is probably the best way to confirm effective starvation. I don't know if there is a way to get a chronic starvation in cells, for sure is possible in animals, but I have some doubts about the possibility to keep starved cells alive for more than 48-72hrs.
If this does not answer to your question, you could provide additional details about what you need to do.
Chronic starvation will eventually lead to cell death. As mentioned by others serum starvation or low glucose/growth factor less media can be used to starve cells for a fixed period of time. These conditions are required when one does not want cells to proliferate e.g., in cell migration studies. As an example I would say that if one intends for chronic starvation of a particular cell line and starts from 10% serum in cell culture and then eventually makes the starvation conditions harsher by going to 5%, then to 2% followed by 1% and finally 0% over the course of culturing. At each stage one would see that more stringent the starvation condition, more would be there cellular death. So overall at the end of chronic starvation one would be left with no cell population to work with.
The starvation should normally take place by gradually reducing the percentage of the serum in the medium. I do not know the type of the cells and the exact type of assay that you want to run with these cells after having them in starvation, but what I normally do is let the cells grow with the proper conditions (e.g. for my cells that is 10% of FCS serum) and when I get healthy, grown-up cells that are confluent on my cell culture flask the reduce the medium to 5% then 2% and finally to 0% of serum. What is normally expected to happen is that your cells will start proliferating slower until they stop proliferating completely. After having your cells in 0% medium, cellular death will occur. It depends on the type of your cells how long the can survive on lower or zero nutrition conditions.
Additional information about the type of the cells and the assays that you aim to perform with them would be welcomed so as a more specific answer for your question to be provided .