Inside, remember even in an artificial gravity (do you have artificial gravity?)
The ship spins like a centrifuge. You can walk around the outside like it's normal gravity but in the centre of the ship (the point around which the rest of the ship's spinning) you're weightless.
In the film of the Martian, the Hermes spins like this.
people probably have to exercise, and without it, they absolutely do; space sickness, infighting, psychological issues, some dire illness, though I assume the ship has a medical bay, what to do with the children on board, or potential ones on board... it's kind of endless both areas.
Astronauts have to exercise. Zero gravity or micro gravity (i.e. weak gravity) plays havoc with musculo-skeletal systems - mostly from muscle wastage and the nervous system adapting to zero gravity. You're not using any of your normal systems (muscles, nerves etc) that keep you upright on Earth. You slowly lose unused muscles. Astronauts who return home after being in space have to readjust to gravity. This usually also involves physiotherapy to strengthen the muscles again.
Bone density's another issue. Even if you create gravity by making the spaceship into a giant centrifuge (they will do this, because the health implicatations of not doing it are too great) if the gravity is less than 1G (i.e. Earth's gravity) people will suffer some degree of muscle and bone wastage. The weaker the gravity, the greater the problem. The international space station has an inbuilt gym and astronauts are required to do regular resistance training to prevent muscle wastage. Even with doing this, they still have to adapt to Earth's gravity again when they return. They're not spending all day every day trying to stay upright. We don't even notice that we're doing that here on Earth (like you don't even notice you're breathing), because the gravity's always there. It's only noticeable if you take gravity away.
If you assume that the spaceship is big enough and spins fast enough to create 1G gravity, then none of the above will be an issue.
Back to the OP's question:
I wouldn't go down the hard science fiction route of having complicated mechanical stuff going wrong with the vessel unless you're very clued up about physics, chemistry and biology* and can get all the details right. You might get away with some very basic things going wrong that requires some basic science knowledge, but you'd still have to get that right. Like if a spaceship runs out of fuel in deep space, it's not going to slow down and stop. It's going to continue at the same velocity (speed and direction) until another force acts on it. As it has no fuel, it can't change its velocity. It doesn't need fuel for most of its journey. It needs fuel to get to the required velocity, then more fuel to change velocity when necessary. But even getting those details right - how do they fix the problem without physics or chemistry? Being handwavey about science for background info (like how the spaceship gets from A to B in its normal, planned route) is fine, but being handwavey about science when it comes to getting characters out of peril is deus ex machina.
*how much I love Andy Weir because he cares about getting the biology right as well, not just doing phys and chem and fudging the biology. And botany as a major plot point.
That leaves all the various human factors that can go wrong... loads of humans in an enclosed space with limited supplies, on a budget, for a period of two years. What could possibly go wrong?
Loads of great ideas on this thread already for various human factors that could go wrong. Think how stressful it is if you're on a plane that's delayed for a couple of hours after boarding, or after landing but before you can get off. Or even just the stress of being on a long haul flight, even without delays. Now put that into context of being stuck on a spacecraft, in the spacecraft equivalent of economy, for two years. Already there's tension. Now factor in stuff that can go wrong...