The answer requires thinking about what you mean by "acceleration".
In SR, there are multiple conceptions of time. When you define $\vec{a} = \frac{d\vec{v}}{dt}$, you must specify whose $dt$. There is a difference between proper acceleration, where the derivative is taken with respect to proper time $d\tau$ (measured on the astronaut's clock), and coordinate acceleration where they are taken with respect to coordinate time dt (measured on a stationary observer's clock).
Due to the commutativity of derivatives, you could also define proper acceleration as the derivative of proper velocity with respect to coordinate time. Proper acceleration is:
$$ \vec{A} = \frac{d}{dt}\left[\frac{d\vec{x}}{d\tau}\right] = \frac{d}{d\tau}\left[\frac{d\vec{x}}{dt}\right], $$
and coordinate acceleration is:
$$ \vec{a} = \frac{d^2\vec{x}}{dt^2}.$$
constant coordinate acceleration?
One consequence of SR is that nothing can be accelerated to the speed of light. Let's say we remain stationary. The astronaut's rocket ship supplies a constant force, which causes the astronaut to accelerate away at $10g$ starting from rest. In SR this won't cause a constant coordinate acceleration. Newton's second law in SR (for a force parallel to the motion, like the rocket) is:
$$ \vec{F} = \gamma^3 m \vec{a}, $$
where $\gamma = (1-v^2/c^2)^{-1/2}$ is the Lorentz factor. As the astronaut's speed increases, so does $\gamma$, and the coordinate acceleration $a$ decreases. In fact, as $v\rightarrow c$, $a\rightarrow 0$.
For the astronaut's coordinate acceleration to remain constant, the force of the rocket would need to increase as time passed. Eventually, it would require infinite force to maintain the $10g$ acceleration. This is why nothing with mass can be accelerated to the speed of light.
In SR no coordinate acceleration can be uniformly maintained for an arbitrary amount of time.
constant proper acceleration
The astronaut experiences bodily stress according to their proper acceleration. The equivalence principle states that a uniform acceleration is indistinguishable from a uniform gravitational field. If the rocket has a proper acceleration of $10g$, then the astronaut feels like they are in a $10g$ gravitational field. If the rocket had a $1g$ proper acceleration, the astronaut would feel quite at home in the $1g$ gravitational field.
The constant force rocket will result in constant proper acceleration:
$$\vec{F} = m\vec{A}$$
Because proper acceleration takes a derivative using the astronaut's clock, time dilation is in some sense the cause of the difference. However, the astronaut does not experience any local time dilation or length contraction. By all experiments the astronaut could devise, they will conclude that they are at rest in a uniform $10g$ gravitational field. All the local lengths inside their spaceship will remain the same, and their clock will happily tick along normally. SR effects are noticed when multiple observers compare measurements in different reference frames. The astronaut would conclude that the Earth is accelerating away, and as it goes faster the clocks on Earth slow down more and more compared to their own perfectly normal clock.
