The SMART error is an automated diagnostic error that detects impending failures in a hard drive. This is indicating your HD is about to fail and needs to be replaced. I believe this is the correct iFixit hard drive replacement guide for your computer. If not, the designs haven't changed much (internally, at least.)
As for data recovery, you can try placing the "bad" HD in an external USB enclosure and seeing if your recovered mac, or another mac, will be able to read it. If not, you may need to seek out a professional data recovery service. Seagate does offer data recovery services.
Let me know if you have any follow-up questions.