Any object moving through spacetime traces out a line called a world line. Even a stationary object traces out a world line because it is moving in time even if it isn't moving in space. General relativity tells us how to calculate these world lines, that is given some spacetime curvature and the initial position and speed on some object we can use GR to calculate the world line of the object.
We can use GR (with a couple of simplifying assumptions) to calculate the spacetime curvature of our universe, and then we can calculate the world lines of objects moving in the universe. We can calculate forward in time to find out how our objects will move in the future, and we can calculate backwards in time to find out how our object moved in the past. It's these calculations that tell us distant objects are moving away from each other i.e. the universe is expanding.
But if we calculate backwards in time then when we reach a time about $13.8$ billion years ago we find that quantities we use in the calculation, such as the average density of the universe, become infinite and since we can't do arithmetic with infinity we can't calculate back any farther. So there is a boundary $13.8$ billion years ago that we can't get past. This boundary is of course the Big Bang and is normally taken to be the beginning of the universe. When we trace back the past world lines of objects they come to an abrupt end at the Big Bang, which is technically described as being geodesic incompleteness.
This is what Hawking means by a boundary i.e. when we trace back world lines they reach the boundary (i.e. the Big Bang) and just end there. However the theory we use to predict this behaviour, general relativity, is a classical theory that does not take quantum mechanics into account. We have no theory of quantum gravity so we simply don't know what really happens at the Big Bang. However 50 years ago two physicists, John Wheeler and Bryce DeWitt, wrote down an equation for a quantum universe that is now called the Wheeler-de Witt equation in their honour, and some time later (I'm not sure exactly when) Stephen Hawking and a co-worker James Hartle found a solution to the Wheeler-DeWitt equation called the Hartle-Hawking state.
The Hartle-Hawking state cannot represent the beginning of our universe for some technical reasons, but it showed than in principle at least a quantum gravity theory that described the beginning of the universe could exist. And better still, when you trace world lines back towards the Hartle-Hawking Big Bang you find they do not end abruptly but instead reach a region where the time and spatial dimensions are ill defined and continue smoothly through that region back out into normal spacetime. So the Hartle-Hawking has no boundary in the way that classical general relativity predicts. The addition of quantum effects has removed the boundary.
This is what Hawking means by there being no boundary. It does not mean time continues on infinitely into the past, and it does not mean that time is looped round like circles on a globe. It is a somewhat technical property that makes the spacetime geometry better behaved at the Big Bang.
Your last question asks if this is the generally accepted view, and the answer has to be no because there is no generally accepted view. The Hartle-Hawking state cannot describe our universe because it applies only to a closed universe and our universe is (we believe) open. Hawking and Neil Turok have attempted to extend the Hawking state to an open universe and the result is the Hawking-Turok instanton. However this is a very strange object indeed, and not everyone is convinced it even makes sense let alone describes the beginning of the universe.
Right now we simply don't know how to describe the Big Bang without the problems of geodesic incompleteness and a boundary. The hope is that the various attempts like String Theory and Loop Quantum Gravity will eventually mature into a full theory of quantum gravity and explain the Big Bang. However until that happens we can only speculate.
