"Untreatable" in this context is used to mean "resistant to all available antibiotics".
Quotes from some recent reviews, editorials, and research papers:
Unemo, M., & Nicholas, R. A. (2012). Emergence of multidrug-resistant, extensively drug-resistant and untreatable gonorrhea. Future microbiology, 7(12), 1401-1422.
The new superbug Neisseria gonorrhoeae has retained resistance to antimicrobials previously recommended for first-line treatment and has now demonstrated its capacity to develop resistance to the extended-spectrum cephalosporin, ceftriaxone, the last remaining option for first-line empiric treatment of gonorrhea. An era of untreatable gonorrhea may be approaching, which represents an exceedingly serious public health problem.
This is saying that gonorrhea strains now exist that are resistant to every one of the proven first-line treatments for gonorrhea. That doesn't mean that it's already untreatable, but when first-line treatment fails it's necessary to go to other backup drugs, including those that typically come with worse side effects or require injections. Once there is resistance to the backups, there's nothing left. Just a couple years before was a paper recommending the switch to drugs like ceftriaxone:
Deguchi, T., Nakane, K., Yasuda, M., & Maeda, S. I. (2010). Emergence and spread of drug resistant Neisseria gonorrhoeae. The Journal of urology, 184(3), 851-858.
Due to the spread of fluoroquinolone resistant N. gonorrhoeae fluoroquinolones are no longer recommended for the treatment of gonorrhea. The emergence of N. gonorrhoeae with a mosaic penicillin-binding protein 2 associated with oral cephalosporin resistance has threatened cefixime treatment for gonorrhea. Emergence of N. gonorrhoeae with high level resistance to azithromycin has also been documented. However, injectable antibiotics (sepctinomycin and ceftriaxone) retain their activity against N. gonorrhoeae
Low, N., Unemo, M., Skov Jensen, J., Breuer, J., & Stephenson, J. M. (2014). Molecular diagnostics for gonorrhoea: implications for antimicrobial resistance and the threat of untreatable gonorrhoea. PLoS medicine, 11(2), e1001598.
Neisseria gonorrhoeae susceptibility to extended spectrum cephalosporins is decreasing and treatment failures are spreading, but no new drug class is licensed to replace them for immediate treatment.
No new drugs are available to treat gonorrhea and the existing drugs are becoming less effective.
Barbee, L. A. (2014). Preparing for an era of untreatable gonorrhea. Current opinion in infectious diseases, 27(3), 282.
The proportion of Neisseria gonorrhoeae isolates with reduced susceptibility to extended-spectrum cephalosporins (ESC) has increased rapidly since 2006. Clinicians, researchers and public health officials need to be prepared for the possibility of an era of untreatable gonorrhea.
Ndowa, F., Lusti-Narasimhan, M., & Unemo, M. (2012). The serious threat of multidrug-resistant and untreatable gonorrhoea: the pressing need for global action to control the spread of antimicrobial resistance, and mitigate the impact on sexual and reproductive health. Sexually transmitted infections, 88(5), 317-318.
Worryingly, resistance and treatment failures to cefixime have been verified in Japan6 and recently in Europe.7–9 The recent report of a strain of N gonorrhoeae in Japan that was highly resistant to the parenteral ceftriaxone, and associated with a probable treatment failure with ceftriaxone,10 the last remaining option for empiric treatment, sounded alarm bells of significant future challenges to the treatment and control of gonococcal infections and their complications. This was also followed by the identification of a highly ceftriaxone-resistant strain in France11 and in Spain.12 Furthermore, many regions, worldwide, describe a decreasing susceptibility to ceftriaxone, and reports of ceftriaxone treatment failure of gonococcal pharyngeal infections have been published.3 ,13 w2 w3 Given the ability of N gonorrhoeae to develop AMR within a relatively short time span compounded with its ability to retain the resistance to previous antibiotics, even after their use has been discontinued, the threat of a widespread ceftriaxone resistance and untreatable gonorrhoea in certain circumstances is real.3 ,10 ,11 w2 w3
The statement itself is not really measurable; "on the brink of becoming untreatable" doesn't have any clear definition (how far away do you have to be to be "on the brink"? how many cases need to be untreatable - one? 5%? 50%?). However, it's certainly a sentiment expressed seriously in scientific and public health venues. Multidrug-resistant strains exist already, there are no replacements in the pipeline, and even if replacements arrive, these particular bacteria have shown high potential for resistance, so it's unlikely that new drugs will remain effective indefinitely.