It could be a lot of things... from the power supply to the motherboard.
My bet its the PSU, but we really have no way of knowing over the internet. I would suggest you find a local PC repair shop--they will be cheaper than taking it to a major business.
Someone that has PC parts lying around could probably try a different PSU and let you know if thats the problem.... (although i must admit that I recently gave up on a PC much like this after trying a different PSU changed nothing)
(Also this probably shouldnt have been in the graphic section.)