assuing you want it working, not just wanting the data

take it 1 step at a time.
Determine where its failing. 
Does it even turn on (fans on , making noises)
Is it also an old monitor (is the monitor OK)

Is is failing to POST
Are there any power on beeps , do the keyboard lights come on , if you press the caps lock on kb does that cap LED turn on/off

Dont worry about thermal paste, that wont ever stop it powering up

Do this , simple things first
Visual inspection: look for corrosion around old CMOS batts , Look for buldging/leaking caps . look for mouse pooh (if mice pee'ed on the mb then it may be dead from corrosion)
try unplugging ALL cables except power , incl USB HD's front cables, reset etc. Power on with with paper clip across power on connectors on motherboard
Unplug & reseat the RAM a few times. Try only 1 RAM module etc

try another power supply
remove CMOS batt completely
try without vid card , it should power up & start beeping

Keep in mind it has ZERO value , far easier to buy another ancient PC of trademe & stick the HD in that (should work if similar mb chipsets)