My MacBook Pro won’t turn on, shows me only the installer log
It has the wrong date. There might be an issue with your battery - that you allowed it to discharge completely. If the issue was only temporary, you can try to set time and date in Recovery (Command R). From there, you also have other options like checking your system partition. Then you restart and hopefully all works again.

Also, there might be menu options where you get the error where you can choose the start up disk. The same you can do by starting with the ALT/Option key.

PRAM reset and SMC reset may also help - and then maybe setting the date again in Recovery.

Additionally, it looks like something was uninstalled which the computer looks for at startup. If you start into safe boot (Shift), the computer may start and you can try to undo what you did, or cleanup the remains of something you deleted. Check the forums for launchagents, cache and such to see what you need to delete. Also check the time and date on your computer while at it.

If the issue is caused by a full hard disk (the reinstall error points to that), delete some unneeded files - cache, downloads and so on, to get enough space. Easiest done by starting from an external disk. If you don't have such a backup-OS, get yourself an empty disk, connect it, start in recovery, and install OS X on it. Then you can start from there and use Finder to make more room on your main disk. That might already resolve the issue.

If that fails, you can now also reinstall the operating system, as you can free enough space on the disk. Before you do that, if you don't have a backup, save the content of your hard disk by copying all the needed data from the system disk. I suggest installing the same version you already have on the system disk, or a newer version. This is easiest from the Recovery mentioned earlier. However, this will not fix all such issues - you might still get the error.

If you do, you have to delete your system disk (never without at least two backups on different disks you are sure to be fine), install the system anew, and restore the last functioning backup or just your data and reinstall your apps.

Edit: There might also be a hardware issue - for instance if you can't access your disk even in Internet Recovery. In that case, you will need a repair before you can use the computer...