Ship Duty Is Work
Monitoring sensors, repairing wear and tear, maintaining life support, plotting course corrections, standing watch, handling alerts, and managing ship systems count as work. A character doing those things is not resting.
Player's handbook
A body can be stationary while a character is still working. Travel aboard a vessel does not automatically count as rest if the crew is actively keeping the ship alive, safe, repaired, hidden, or on course.
Core rule
A character rests only when they are relieved of active responsibility and can sleep, recover, or perform light activity without needing to keep the ship operational.
Monitoring sensors, repairing wear and tear, maintaining life support, plotting course corrections, standing watch, handling alerts, and managing ship systems count as work. A character doing those things is not resting.
A rest during travel requires enough crew coverage, automation, safety, and uninterrupted time for the resting character to be off duty. If the ship cannot safely run without them, they are working.
Rests
Use normal fifth edition rest rules, with shipboard duty determining whether the character actually qualifies.
A short rest can occur during travel if the character is off duty and not performing active ship work. Light conversation, eating, basic personal maintenance, and calm recovery are fine.
A long rest requires enough uninterrupted off-duty time for sleep or equivalent recovery. Routine travel time does not count if the character is responsible for watches, repairs, piloting, sensors, or emergency response.
Combat, emergency repairs, hull breaches, failed life support, navigation crises, alarms, or other urgent duties can interrupt a rest as normal. The DM determines whether the rest is preserved or lost.
Downtime
Travel time that is not rest may still support downtime activity if the ship is stable enough and the character has the right tools, materials, and attention.
Characters may research, craft, repair, train, map, decode, recover information, maintain relationships, or work on ship projects during travel if their duties leave enough usable time.
Dangerous routes, damaged systems, enemy pursuit, low crew coverage, unknown anomalies, or failing life support can prevent downtime even if the trip takes many hours or days.
Exhaustion
The DM may call for a Constitution saving throw to avoid exhaustion when conditions push a character beyond ordinary fatigue.
Overlong watches, too few crew, emergency repairs, repeated alarms, operating a damaged ship, or keeping critical systems alive without relief can risk exhaustion.
Vacuum exposure, radiation, failed life support, toxic air, extreme heat or cold, dehydration, starvation, alien infection, toxins, and dangerous environmental travel can risk exhaustion.
Cryo sickness, jump sickness, neural overload, prolonged acceleration, anomalous space, or other setting-specific strain can risk exhaustion when the fiction makes the stress severe.