Saudi Arabia, along with Egypt, Bahrain and the United Arab Emirates, announced last month that they would cut diplomatic ties with Qatar, accusing it of supporting terrorism, which Qatar vehemently rejected. Sky News reported.
The group cut air, sea and ground links with the major oil exporter, cutting off vital routes for imports, including food.
A list of 13 demands were made on Qatar in order to have sanctions lifted but on Wednesday Qatar rejected the list.
On Friday, the four Arab states said in a joint statement that Qatar's refusal to agree to the demands "reflected its intention to continue its policy, aimed at destabilizing security in the region".
They added: "All political, economic and legal measures will be taken in the manner and at the time deemed appropriate to preserve the four countries' rights, security and stability."
There was no detail on the form these measures might take.
The four countries said Qatar had "thwarted diplomatic efforts" to resolve the crisis, adding that the demands, sent through mediator Kuwait, were now "null and void".
Among the 13 demands made by the Arab states was the closure of Doha-based broadcaster Al Jazeera, along with other media such as the New Arab and the Middle East Eye.
Other key demands included severing ties with groups such as the Muslim Brotherhood and Hezbollah, closing a Turkish military base and downgrading links with Iran.
Qatar has denied any involvement in supporting terrorism and said the Persian Gulf states' demands were so tough that they were never seriously meant to be negotiated.
Earlier this week, Qatari foreign minister Sheikh Mohammed bin Abdulrahman Al Thani, said the charges "were clearly designed to create anti-Qatar sentiment".
He said: "Reading between the lines, the blockading countries were demanding that we have to surrender our sovereignty to end the siege, something which... Qatar will never do."
He continued: "Qatar continues to call for dialogue despite the violation of international laws and regulations, despite the separation of 12,000 families, despite the siege that is a clear aggression and an insult to all international treaties, bodies and jurisdictions."
US secretary of state Rex Tillerson will fly to Kuwait on Monday amid escalating efforts to bring the crisis to an end.
Germany has also offered the help of its intelligence services to clear up allegations of Qatari funding of extremism.