Can a spaceship accelerate at $1g$ forever? Let's say a spaceship is designed to accelerate continuously at 9.8m/s/s, and thereby create an earth-like environment on one surface for its inhabitants to live.
From the perspective of the inhabitants (and their descendants) could this carry on forever? Surely the ship would get closer and closer to the speed of light.... Does this restrict the possibility of creating a 1g environment indefinitely on the ship? Would the inhabitants notice any difference between the slower and more relativistic speeds?
 A: You're describing constant proper acceleration motion and, yes, in principle it is absolutely possible (there might be practical problems though - but they are unrelated to the principles of your question).
You can always accelerate at any rate relative to your instantaneously comoving frame. No physical limit hinders you in any way. By Galileo's principle, this must be true: your physical situation, ability to accelerate and so forth cannot change whatever your relative velocity to any other observer (including those comoving with your motion at any former time).
Imagine the following. You begin in an inertial frame, say frame $A_0$. You boost to a velocity $v$ relative to that frame over a time (as measured by you) to a new inertial state $A_1$. Now you do exactly the same again, with the same physics, to boost to state $A_2$: your situation is exactly the same, and the relationship between frames $A_2$ and $A_1$ must be exactly the same as that between $A_1$ and $A_0$, by Galileo's relativity postulate. Keep making the same jump. By Galileo's principle, the relationship between frames $A_n$ and $A_{n-1}$ must be exactly that between $A_1$ and $A_0$, including the acceleration you feel as you make the boost.
If you like, you can there are intergalactic comoving fuel stations everywhere (by COMOCO Oil TM) so that, in whatever state, you can take on fuel needed for the boost from frame $A_{n-1}$ to frame $A_n$, so that your situation (rest mass and so forth) is exactly the same for each jump.
The universal signalling speed limit $c$ works as follow. The time dilation in each frame $A_n$ relative to frame $A_0$ increases with $n$, so that, from $A_0$'s standpoint, each successive jump adds less speed so that your speed, relative to your initial frame $A_0$, asymptotes to $c$. This is the mechanism that prevents greater than $c$ relative motion between observers: from your point of view, there's no barrier ever to your accelerating indefinitely.

I spoke of practical problems. You're almost certainly going to vaporize yourself eventually if you try to move with speeds of an appreciable fraction of $c$ relative to the intergalactic matter and gas in any region. See Randal Munroe's "Relativistic Baseball" Whatif.
A: Acceleration is a change in velocity, and velocity describes both speed and direction. What that means is that an object (a space ship) can be accelerated in such a way that the direction is constantly changing, but the speed is not. The simplest and most obvious example is going in a circle. This is an answer to the question not just n a technical sense, but in a practical one too. It could be used to provide a gravity-like force, possibly for long term inhabitance by humans (and their pets.)
The force of the fake gravity will depend on position, but to be fair, that's true of earth's gravity too. It's just that the change in gravity with position is so small we don't notice it. It's not at all inconceivable that a space ship could made that is large enough that humans like us could adapt to any difference in sensation compared to earth.
So how would this be implemented? Make a space ship with an engine that causes it to go in circles, doing perpetual donuts in space? That could work, but a more likely solution would be a spaceship shaped like a giant bicycle wheel. If the wheel is spinning then anything inside the tire will be pushed outward. Your traveling community would live on the inside of the tire just opposite the tread. Bigger ships would need to spin more slowly to stay at 1g.
These are hardly my ideas; I'm sure they've been part of science fiction for 100 years.
A: Constant rotational speed (angular acceleration) can provide 1G simulated gravity. There is no need for infinite (linear) acceleration to meet that requirement.

The ship would get closer to the speed of light.... Does this restrict the possibility of creating a 1g environment indefinitely on the ship?
Theoretically you could accelerate indefinitely forward, but at smaller and smaller increments as you approached the universal speed limit of Light. It would also become infinitely more expensive in terms mass-energy to continue to accelerate further, and the increments will diminish below 1G. So the answer is..YES, there is a limit, unless you opt for aforementioned rotational acceleration. You're better off making very expensive espressos on the International Space Station in simulated gravity.
Would the inhabitants notice any difference between the slower and more relativistic speeds? - No, all Observers experience proper time normally regardless of frame of reference, even if they achieved 99.99% speed of Light, or fell into a Black Hole.
