Sounds like that most irritating problem, "the undiagnosed hardware fault". Will probably take some hands on fiddling with to diagnose. Usual method is to strip the PC down to basics, ie one stick of RAM and no peripherals, where hopefully it will work, then one by one put things back until you find the problem component and replace it.
Does the desktop have a graphics card, or does it use on-board MOBO graphics ?
As to rescuing data, this depends on what's gone pear shaped. If the hard drive is OK then the data is safe, if it's died then you have a problem.