The concept is far from obsolete, but the use of the word barratry implies a body of law that for the most has vanished from use.

In most sensible common law jurisdictions being a common barrator was a common law offence; most such jurisdictions have allowed common law offences to fall into disuse, and many have outright abolished effectively all of them in favour of codified criminal statutes.

There is statutory relief available against litigious harassment in most such jurisdictions, either directly in the criminal code, or in the statutes controlling the bodies that regulate the legal profession. Additionally, the codification of rules of practice in litigation also controls what litigants without professionally regulated representation may do without facing their case being struck out on application by the counterparty. There is also often statutory relief against baseless litigation (such as you might find in SLAPP).

The bar for a successful prosecution against egregious litigators is quite high because it is a non-strict-liability criminal offence; you have to prove intent as well as action. For single isolated cases that may have some surface merit, it is easier to get the trial court to strike out the case, or to arrive at an early judgement with costs consequences. If the prima facie vexatious party continues litigation courts are generally empowered to consider indemnity orders on costs (which turns on accelerated interest on all the costs payable by that party) and to require costs deposits (where the prima facie vexatious party must leave with the court the costs that will cover the litigation expenses of the other party on the grounds that an adverse costs order is probable), and finally courts are also generally empowered to make interim costs orders in especially bad cases, wherein the prima facie vexatious party must pay some of the other party's costs before final judgment (this is to avoid a rich vexatious litigator from forcing an adverse settlement upon a party that is likely to ultimately prevail).

Finally, there is usually a modern distinction drawn between a compulsive claimant (who is probably mentally ill) and a vexatious one (who usually targets one or a small group of defendants).

The law in England and Wales offer several means of dealing with both.

As one example, the Civil Rules of Procedure rule 3.1 and Practice Direction 3A allow a district court to impose a civil restraint order on litigants that tightly controls what they may do in the remainder of the current proceedings, or for a period of time in any district court, or generally in the courts of England and Wales. CROs require people served with them to apply in writing to a court before making various applications or claims, and if permission is not obtained, the application or claim is automatically dismissed (with costs consequences).

In issuing CROs courts will generally consider, among other things, the tendency of a subject to engage in frequent litigation, the manner in which litigation is undertaken, the subject's responsiveness to the courts and the other parties during the course of the litigation, whether litigation is being needlessly and wilfully prolonged, and the effect that the litigation has upon the defendant or defendants.

Alternatively, the Attorney-General (of England and Wales), a member of the government, may apply to a High Court for an order under the Senior Courts Act to declare a litigant vexatious. This applies in all courts -- not just district ones -- and may further require a claimant or applicant to seek permission from the Ministry of Justice as well as the Court. A list of people who have had such orders made against them is here: https://www.gov.uk/guidance/vexatious-litigants

Few of these people would have been convictable as barrators under the pre-1967 common law definition; and additionally a few of those on this list whose orders date back before 1967 likely would have been the subjects of district court CROs instead.

Most of them, on the other hand, suffer from intractable -- or worse, untreated -- mental illness. :-(

Finally, bringing a ridiculously large claim in England and Wales cannot be done in the district courts, and the High Courts will generally quickly reject -- with costs consequences -- claims that are on their face too large for the damages particularized in the statement of claim. In England and Wales, district courts are used for claims under £10 000. "Go back to a district court and get on the small claims track as you cannot prove the money value in your claim of £millions" is a very expensive order to receive from a High Court, and that is almost certainly what the claimant in the case in the article would have to overcome.