But the prime minister promised to push for the "greatest possible" access to the single market following Brexit.
In a long-awaited speech, she also announced Parliament would get a vote on the final deal agreed between the UK and the European Union.
And Mrs May promised an end to the UK's "vast contributions" to the EU.
The prime minister used the speech to announce the UK's priorities for Brexit negotiations, including:
Mrs May said there would not be a "blow-by-blow" account of negotiations, set to get under way after Article 50 of the Lisbon Treaty is invoked by the end of March. It was not her intention to "undermine" the EU or the single market, she added.
But she warned the EU against a "punitive" reaction to Brexit, as it would mean "calamitous self-harm for the countries of Europe and it would not be the act of a friend".
She said: "This agreement should allow for the freest possible trade in goods and services between Britain and the EU's member states.
"It should give British companies the maximum possible freedom to trade with and operate within European markets and let European businesses do the same in Britain. But I want to be clear: what I am proposing cannot mean membership of the single market."
Mrs May added: "It would, to all intents and purposes mean not leaving the EU at all. That is why both sides in the referendum campaign made it clear that a vote to leave the EU would be a vote to leave the single market."
The single market allows the free movement of goods, services and workers between its members.
The government had previously revealed few details about what it wants to secure from the Brexit talks, with Labour urging Mrs May to push for a "deal that works for trade".
EU leaders had warned that the UK cannot "cherry pick" access to the single market while restricting the free movement of people.
Mrs May said: "We will continue to attract the brightest and the best to work or study in Britain - indeed, openness to international talent must remain one of this country's most distinctive assets - but that process must be managed properly so that our immigration system serves the national interest.
"So we will get control of the number of people coming to Britain from the EU."
Addressing an audience including senior ministers and foreign ambassadors in central London, Mrs May said the UK had "voted for a brighter future for our country" and would become "stronger, fairer, more united" after Brexit. She said the country's history was "profoundly internationalist" and would remain so.
The prime minister said the UK had often been seen as "an awkward member state", but the EU had not demonstrated "enough flexibility on many important matters for a majority of British voters".
She told the remaining 27 EU member states: "We will continue to be reliable partners, willing allies and close friends. We want to buy your goods, sell you ours, trade with you as freely as possible, and work with one another to make sure we are all safer, more secure and more prosperous through continued friendship."
Mrs May, who backed Remain in the referendum, called for a "new and equal partnership" with the EU, "not partial membership of the European Union, associate membership of the European Union, or anything that leaves us half-in, half-out".
"We do not seek to adopt a model already enjoyed by other countries. We do not seek to hold on to bits of membership as we leave."
People who voted Brexit "did so with their eyes open", the prime minister said, calling the vote a "great moment of national change".
She said the country was "coming together" after June's referendum, adding: "Now we need to put an end to the division and the language associated with it - Leaver and Remainer and all the accompanying insults - and unite to make a success of Brexit and build a truly global Britain."
Since becoming prime minister last July, Mrs May has faced repeated calls to provide more detail on her Brexit strategy, but has refused to offer a "running commentary", saying to do so would weaken the UK's negotiating hand.Labour's Sir Keir Starmer, the shadow Brexit secretary, said the UK should stay in the customs union, telling the BBC: "We have to decide how we front up to these negotiations. We need to aim for the best deal for the UK.
"That's the deal that works for trade. It's a deal that accepts there has to be change on freedom-of-movement rules. That's the starting position.
"Preserving our ability to trade successfully in Europe has to be the priority for business. Staying in the customs union is the best way to achieve that."
UK voters opted for Brexit in last June's referendum by 51.9% to 48.1%.
After Mrs May's speach Liberal Democrat leader Tim Farron said: "Hard Brexit was never on the ballot paper. Ripping us out of the single market was not something proposed to the British people. This is a theft of democracy."
In a statement, the Irish government said the UK's "approach is now firmly that of a country which will have left the EU but which seeks to negotiate a new, close relationship with it". It added it was "acutely aware of the potential risks and challenges for the Irish economy" but also of "the potential economic opportunities that may arise".
Earlier, German Chancellor Angela Merkel urged the business community in her country to back her stance on Brexit, saying: "I'm asking you as representatives of the business world to act together with us, because should it become apparent that you can get full access to the single market even if you can choose certain things then we risk that every country cherry picks. That's why politics and business need to act together."