tl;dr: Yes, the paper provides strong evidence that we're undergoing a mass extinction event. The media reporting is basically right (for once).
What you are asking
I'm going to assume that what you're asking amounts to: "Is there any obvious flaw in the methodology of the recent study on this topic, that the (notoriously poor) mainstream science journalists have overlooked?". The reason I'm making this assumption is because I think that, if the claims of the study are as reported, the claim mentioned in the question must follow. If for some reason you're not convinced by the scientific method in general, then I doubt we can help you here.
The actual findings of the study
Okay, so first let's look at the study, not the bbc news article. It's authored by reasonable looking academics, at well known and important institutions. It's published in a good place. Because of these two things, I'm inclined to take whatever they actually say in the study at face value, so let's look at what it says.
If you read the study, you find that the authors define a mass extinction to be a time period when the rate at which species go extinct is much higher than the "background rate". The rate that species go extinct is expressed in somewhat funny sounding units, but basically it's the fraction of species that go extinct each year. That is, the number of species that went extinct, divided by the total number of living species at the start of that year. This lets us sensibly compare periods of high and low biodiversity.
The authors use what is described as a "very conservative" estimate for how often species go extinct during normal life (i.e. not during a mass extinction event). This estimate is arrived at by rounding up the latest best estimate of how often mammals have gone extinct (derived from the fossil record), and then assuming that all types of organisms go extinct that fast, even though we have good reasons to think that this is untrue (apparently invertebrate species live longer?). The authors overall find that for every 1,000,000 species you have alive at the start of a year, about 2 will be extinct by the end of the year. This means that a typical (mammalian) species will live for about 500,000 years.
The authors then look at the records of species we know about, and count how many of those have become extinct in the last century. They find that far more than expected have become extinct. About 8-20 times as many using the "very conservative estimate", and more like 50-100 times as many using a less conservative estimate. This handy graph gives you a pretty good image of what they find. The dashed "Background" line at the bottom shows how many species should have gone extinct, while the other lines show how many actually did.
Okay, so what can we conclude from the paper? Well, taken at face value, it looks like there's a mass extinction in progress. Some arguments against this conclusion might be:
The paper doesn't actually offer a statistical analysis of this phenomenon. I think this is a little sloppy, even though the effect is so large that it almost doesn't matter. Basically they need to look at the variance in the baseline extinction rate, and give us a range of normal values. However, I rather suspect that the range won't be 10-20 times larger than the mean value.
The paper relies on observed extinctions. It might be the case that lots of species go extinct all the time, and we just haven't gotten very good at keeping track of things like that until now. I think this is a valid concern. It could be that there are massive and systematic biases in the fossil record, and that species are dying off in large numbers all the time. However, the shape of the curve the authors present suggests otherwise. If species were dying all the time, then certainly after about the 1850s, when citizen science got big and people were actually doing a good job of keeping track of numbers of different species, we should expect the extinction rate to be basically flat. It shouldn't be increasing if this is the usual rate stuff dies at.
- In the comments, @matt_black and others raise a criticism of the paper on the basis of its authorship. I didn't realize this was a controversial issue when I started writing the answer, so I didn't investigate it then, but have now.
In particular, the second author, Paul Ehrlich, has made many widely publicized and incorrect predictions. Most notable was the prediction that hundreds of millions of Indians would die of starvation during the 1970's and 1980s (Counterpoint: Ehrlich is quoted in the given Wikipedia link in the 1990's claiming that the statement in the book was an example, not a prediction, and that the core message of the book was still basically correct). As such, some people would be inclined to discount the paper's findings because its author has a history of publishing somewhat sensationalist and incorrect predictions. As a counterpoint, we can note that this is a logical fallacy: if the paper's methodology is sound, then it doesn't matter who is making the claims. However, going a step further, the paper's opening sentences are:
The loss of biodiversity is one of the most critical current
environmental problems, threatening valuable ecosystem services and
human well-being (1–7). A growing body of evidence indicates that
current species extinction rates are higher than the pre-human
background rate (8–15)
Examining the cited references, we find that they include a number of papers that I would characterize as having a high, apparently positive, impact on the scientific community. For the first claim (That the loss of biodiversity is a big problem), we see this paper, which boasts nearly 600 citations over 12 years. I did not look at every instance of it being cited, but a sample of 10 citing papers I selected at random contained no citations I would characterize as negative, being roughly evenly split being those I would characterize as endorsing the findings and those I would characterize as using the manuscript to support specific factual claims in their own paper.
For the second claim, the most cited of the 7 listed papers is this one, which I would again characterize as having an extremely high impact on the scientific community, which appears to be largely positive. The paper's opening lines make the claim that the extinction rate is 100 to 1,000 times larger than the estimated baseline rate, which is actually a much more outlandish claim than the one made in the paper we're discussing in this question (about an order of magnitude larger). The paper has more than 1800 citations over the last 20 years. Because the paper is older, and scientific consensus changes over time, I looked at the first two pages of google scholar's list of papers that cited this one in the last 2 years, and checked all the papers that were not paywalled. Included among them is a second study by some of the same authors from the last two years, which again claims (without citation) an extinction rate 1,000 times the baseline. So at least we know the original authors still endorse this view. Of the remainder, I find 4 that restate the key claim (100-1,000 times the baseline rate) with citation, 5 that cite it to support a somewhat watered down version of the claim (diversity is declining "fast"; Humans are "dramatically altering biodiversity around the globe"; Human actions have "generally negative" impact on biodiversity; Ample evidence that extinction rates are increasing), 3 that cite the paper in support of another, more minor claim (diversity is decreasing in specific regions; certain factors making extinction more likely; reasons for certain fauna being rare in Britain; there are lots of species of fungus we don't know about yet), 1 that does not cite the paper at all, 1 that is the paper we are discussing, and 2 self citations.
To summarize, I found 9 papers citing the core claim as fact (with more or less hedging of the wording); 3 citing it for other reasons, and 3 miscounts by Google Scholar. Overall, while my sample of 15 papers is small, I'd characterize this as a claim that is probably broadly accepted. If there is controversy, it seems to be well hidden.