You can't do a backup if you can't power it on. But if you can manage to power it on just for a few minutes, the first thing to do is copy out your saved data to USB.
Google for the flashing blue light, there are some basic steps like holding down the power button to get into safe mode etc. Try those first.
I would also recommend examining the PS4 yourself. Open it up to look for obvious things like dust buildup, clean up the dust, especially from the fan. You will need a special screwdriver though (T9 torx I think, can buy from hardware/DIY stores). Look up some youtube videos. It's relatively safe, just open the casing to expose the interiors and the fan, not dismantling any more than that, no wires to deal with.
If have to send for repair, remove the HDD. Surely the Sony repair center has some spare HDD they can use to examine the ps4, just tell them you already removed the HDD. Otherwise put in any spare hdd you may have. If their repair doesn't replaces the motherboard, there's a chance your old HDD (removed from the ps4) can still work after the repair. After they investigate, before they attempt anything major, they should call you to give you an estimate of the cost anyway (or you can remind them to do so), ask them what they will change and whether your removed HDD will still work (ask to speak to the engineer doing the repair, if the person on the phone doesn't know). [ps. I have not dealt with ps4 repair center before. The above is just general repair center process.]
If the account that was used to play the games is PS+ member, it's likely that the game saves were already uploaded to the cloud. (This requires certain settings to automate the regular upload, as well as putting the PS4 in rest mode instead of powering off completely.) If cloud backup was automated, then the game saves are still recoverable.