This is usually a power supply or motherboard failure.
To rule out other things, disconnect all other cards
including video, pull memory out, disconnect hard drives too
from both the board and PSU.
Having only CPU + heatsink/fan installed, try to turn the
system on again. It should beep because there's no video
or memory.
If it beeps, try clearing CMOS and put the video and memory
back in. If it doesn't, observe whether the PSU fan stays
spinning, LEDs lit besides the on on the mainboard.
You can try powering on just the PSU by unplugging it from
everything but one hard drive, then short the green PS-On
pin to ground with a paperclip (assuming it is a standard
ATX PSU). At that point the hard drive and PSU fan should
spin and if you've a multimeter you can take voltage
readings. If it won't turn on, this indicates the PSU has
failed. If it does turn on and the voltages look good, you
can either examine the PSU interior (with it unplugged,
capacitor failure is a common problem which is usually
visible from venting), swap in a different PSU, or begin to
suspect the motherboard.