At least once I had the appropriate tools at hand ..

tl;dr: How to retrieve ext4-data from the broken laptop’s drive with a Mac mini (OSX 10.9)
* download and install FUSE for OSX
* download and install fuse-ext2
* reboot
* mount the specific partitions via diskutil & copy the wanted data & wait ..

OK, the situation is: from one day to the next my laptop stopped starting. Power it on, but no screen, immediate shut-off. Yes, I could spend now hours and hours to check for the issue, but frankly said: the device is six years old, was always efficient for software development, texting and browsing and simple photo-editing, but now its time has come.
But I need the data?
Yes, I have some backups. The newest one is 1 month old, so the most interesting data is not retrieveable. Therefore I opened the device with a set of smartphone-tools, fetched the SATA-drive, plugged it USB2.0-to-SATA-drive connector and plugged it into the Mac mini. The drive was immediately shown, but none of the fice partitions were mounted. Two were NTFS, two ext4 and one Linux-swap. I was just interested in the Win7-partition and the ext4-home.
So I installed the two aforementioned tools et voila …

opened back of the laptop