No, definitely not to the degree that the Buzzfeed article implies. Those videos are from research by Jack Gallant at Berkeley, and are showing estimates of what a subject is currently watching in an fMRI scanner. Essentially they use a large amount of training data to build a model of the what visual features activate different brain locations, then try to work backwards from brain activity to the most likely video the subject is watching. Note that they construct these most-likely videos by averaging together YouTube clips, so some of the structures you see in the videos are artifacts from the input videos.
There has been work from Berkeley as well as other labs on trying to decode memories, which has had some success but is far from a video recording - for example, they can predict above chance which of 5 pictures a subject is thinking about. Using data from current or near-future brain imaging technology, it is unlikely that any analysis method will be able to reconstruct videos from memories.
This Wired article, "Scientists Can’t Read Your Mind With Brain Scans (Yet)", has a decent summary of the current state of the field:
Most “mind-reading” studies so far have focused on the here and now. “Vision is by far the easiest thing to work with,” said Jack Gallant, a neuroscientist at the University of California, Berkeley. Gallant’s lab has done some of the most eye-catching work in this area, including a 2011 study that used fMRI scans to decode video imagery as people watched clips cut from Hollywood films (see video below).
They're unable to read memories yet:
If decoding what people see and what they’re just about to do next is where the field is now, and decoding mental imagery is what’s on the horizon, Gallant says there’s yet another type of decoding that’s more like the distant frontier: decoding old memories. “If I ask you to picture your first grade teacher, you might be able to recall his or her name and call up a pretty rough mental image,” he said. “That information is buried in your brain, but you probably hadn’t thought about it for years,” Gallant said. Scientists don’t understand how old memories are encoded in our brains well enough to decode them, but some day they might.
If you’re starting to feel a panic attack coming on, take a deep breath and relax. Scientists are very, very far from being able to dredge up those best-forgotten memories from your grade school days (or worse, junior high). They don’t even want to.