She was born in Orkdal Municipality in Sør-Trøndelag county to Terje and Berit Melby, a bank clerk and school assistant.

[7] In 2019, she and Petter Eide from the Socialist Left Party, demonstrated at Eidsvolls plass during the Norwegian reception of Li Zhanshu, the chairman of the National People's Congress's Standing Committee.

Melby and Eide were asked to take off their yellow T-shirts with the inscription "freedom" in Norwegian and Mandarin before they could enter the Storting.

Melby stated afterwards that it was "at best naive of the Storting to accept that our democratic rules are deceived by a manipulative system, at worst we have allowed ourselves to be dictated, and that is even worse".

She was criticised by many educational experts for her decision, but she defended it by saying that it would be up to the schools respectively to decide and notify their county governor about it.

The Conservative's Turid Kristensen also criticised Melby, stating that they didn't want to build down common subjects.

She expressed that "the Labour Party now has the opportunity to show that they support ordinary people, and that it’s not just empty words and rhetoric.

[21] In November, after minister of culture Anette Trettebergstuen stated that no members of the government or representatives of the state would attend the 2022 FIFA World Cup in Qatar, Melby praised Trettebergstuen, while adding that no members of the Storting should attend the event either.

[22] In June 2022, Melby reiterated the Ukrainian ambassador's sentiment that the Norwegian government should be harsher against Russia.

The fact that the government is undermining the sanctions to prevent the economic loss from becoming too great for us only contributes to financing Putin's war machine even further.

Putin removes the small glimmer of hope for a negotiated solution, and warns that the war will last a long time.

[27] On 19 November 2023, Melby declared her candidacy for Prime Minister in an attempt to appeal to voters from the right wing-bloc who were dissatisfied with Erna Solberg's intentions of running for the post again in the 2025 election.

[28] The Green Party leader Arild Hermstad expressed support for Melby's candidacy, calling it "a good idea".