There has been some discussion of what happens when an extended object
undergoes uniform acceleration.
-- Relativity issues arise for long objects and/or large accelerations,
i.e. when L*a is not small compared to c^2.
-- Different (simpler) relativity issues arise when the duration of the
maneuver is such that a*t is not small compared to c.
The result is known as /hyperbolic motion/, for reasons that are obvious
if you look at the spacetime diagram. Note that a hyperbola is to
Minkowski space as a circle is to plain old Euclidean space. It's
pretty fundamental.
Note that the result can be quite surprising if you don't think about
it in just the right way. This is one of those situations where a
simple question leads to a rather intricate answer. Here is my
favorite approach: https://www.av8n.com/physics/hyperbolic-motion.htm
With a bit of cleverness, you can engineer it so that the object keeps
its shape during the maneuver. This can be described using some rather
simple equations, along with some nifty diagrams.