The PM promised to push for the "freest possible trade" with European countries and warned the EU that to try to "punish" the UK would be "an act of calamitous self-harm".
She also said Parliament would vote on the final deal that is agreed.
Labour warned of "enormous dangers" in the prime minister's plans.
And the European Parliament's lead negotiator said there could be no "cherry-picking" by the UK in the talks.
Mrs May used her much-anticipated speech to announce her priorities for Brexit negotiations, including maintaining the common travel area between the UK and Irish Republic and "control" of migration between the UK and the EU.
Negotiations are set to begin after notice under Article 50 of the Lisbon Treaty is served by the end of March.
It was not her intention to "undermine" the EU or the single market, Mrs May said, but she warned against a "punitive" reaction to Brexit, as it would bring "calamitous self-harm for the countries of Europe and it would not be the act of a friend".
She added: "I am equally clear that no deal for Britain is better than a bad deal for Britain."
Since the referendum she and her ministers have simply refused to be so explicit.
For months, some ministers have privately whispered about complex solutions that might keep elements of membership - the choices not being binary, mechanisms that might give a sort of membership with a different name.
Well, no more. The simple and clear message from Theresa May's speech this morning is that we are out.
The prime minister had some strong words of advice for the EU and its treatment of member states, arguing it could "hold things together by force, tightening a vice-like grip that ends up crushing into tiny pieces the very things you want to protect" or "respect difference, cherish it even".
But the most keenly awaited part of the speech dealt with the UK's post-Brexit trading relationship with the rest of Europe.
Any agreement with the EU must "allow for the freest possible trade in goods and services", Mrs May said.
"But I want to be clear: what I am proposing cannot mean membership of the single market.
"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."
EU leaders have warned that the UK cannot access the single market, which allows the free movement of goods, services and workers between its members, while at the same time restricting the free movement of people - and the PM has pledged to control EU migration.
Mrs May also indicated the UK's relationship with the customs union - under which EU countries do not impose tariffs on each other's goods, while all imposing the same tariff on goods imported from outside the EU - would change.
She said she did not want the country to be "bound" by the shared external tariffs. Instead, the UK would be "striking our own comprehensive trade agreements with other countries".
To the 27 other EU member states, she said: "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."
When asked about the prime minister's promise of a Parliamentary vote following Brexit negotiations, her spokeswoman said: "You can regard it as binding."
Pressed on what would happen if MPs or peers rejected any deal, she replied: "Either way, we will very clearly be leaving the EU."
Until now, Mrs May had revealed little of her strategy for the talks, which could last up to two years - or go on longer if all 28 EU members think this is necessary.
Responding on Twitter, Guy Verhofstadt, the European Parliament's chief negotiator, welcomed Mrs May's "clarity", adding: "But the days of UK cherry-picking and Europe a la carte are over."
In a reference to Mrs May's warning that the UK could "change the basis of Britain's economic model" if denied single market access - taken to mean lowering corporation tax to attract businesses - he added: "Threatening to turn the UK into a deregulated tax heaven will not only hurt British people - it is a counter-productive negotiating tactic."
Mr Verhofstadt added that the views of people who voted Remain must be taken on board.
Labour leader Jeremy Corbyn argued that the prime minister still needed to "be clearer" about her long-term objectives, and that she she wanted to "have her cake and eat it" over the single market.
He added: "I think we have to have a deal that ensures we have access to the market - we have British jobs dependent on that market - that's what we'll be pushing for."
Mr Corbyn also said: "There are enormous dangers in all of this and when she talks about future trade arrangements, all she said was that Donald Trump said we'd be first in the queue - first in the queue for an investor protection-type treaty? I don't know exactly what she has in mind on that."
After Mrs May's speech, Liberal Democrat leader Tim Farron said: "Ripping us out of the single market was not something proposed to the British people. This is a theft of democracy."
UKIP leader Paul Nuttall said he feared a "slow-motion Brexit", adding: "We want this done quickly."
Scottish First Minister Nicola Sturgeon claimed leaving the single market would be "economically catastrophic".
She hinted at a second independence referendum, saying Scotland - which voted against Brexit - should have "the ability to choose between that and a different future".
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".
Michel Barnier, the European Commission's chief Brexit negotiator, tweeted: "Ready as soon as UK is. Only notification (that is, invoking Article 50) can kick off negotiations."
Source: BBC Business News