HELP! Retrieve data from Mac that won't boot!

So I'm regretfully desperate to retrieve date from my Mac (I'm unsure of the model, I think it was made in 2010-11). I realized I was running out of disk space so I began to painstakingly transfer photos to my external hardrive, it was running terribly slow and really wasn't even doing anything, so I turned it off, waited a bit and then turned it back on. At this point it turns on, shows a white screen with the Apple logo in the center like normal, then a circle with a line through it takes the Apple logos place, then a few seconds later the Apple logo appears again along with the small loading bar below. It continues to the disk utility screen with different languages scrolling across the screen. My dad is a computer tech but is better with PC's, so we're trying all we can to retrieve the data that I regretfully didn't back up! We've tried to run the disk repairs but that didn't work, he has tried several other options that have also not worked. It appears the hard drive has a corrupted MBR, this disk is full so we are so unable to reinstall the OS with the partition on it. We are going to try target disk mode with my cousins Mac, but mine has FireWire and his has thunderbolt so I'm having trouble finding a cable that doesn't cost $60 in case this doesn't even solve the problem! If you have any opinions or thoughts on where I should go from here, it would be so appreciated! If you need more information kindly let me know!