MacBook Pro early 2011 hangs at “white screen”
Unfortunately it is almost certainly the discrete GPU on the main logic board that is failing here, not some kind of OS issue. This symptom is the most common way this issue presents in the 2011 models, especially considering the fact that your machine isn't able to boot to recovery, but does not display any boot error messages (like the ? folder or prohibitory symbol). 

The 2011 MacBook Pros had a repair extension program from Apple for this issue free of charge for a couple years. Unfortunately, it ended a while ago. 

You can salvage the data in target disk mode https://support.apple.com/en-us/HT201462 if you have another mac to connect yours to. You could also just remove the hard drive and connect it to another computer with an enclosure to recover data if that's what you need from it.

I know that there are definitely repair shops that do component level repairs on the failing GPU, but it would not be free. That said, Apple no longer fixes the 2011 models at all, even if you were to pay, so an independent repair shop would be your best bet anyway if you really wanted to get this computer fixed.