Partially within an area is within the area
Faerie Fire says:
Each object in a 20-foot cube within range is outlined in [...] light. Any creature in the area [...] is also outlined...
So if the creature or object is in the area, then it is affected, outlined entirely by the faerie fire effect.
Does a creature or object have to be fully in the area to be affected? Definitely not. That isn't how spells are ever expected to work. For example, if an ogre is only partially in the area of a fireball spell, our expectation is that the ogre is affected and takes full damage. No rule suggests that it would take damage based on what percent of its body is inside the spell's area. Conversely, if an ogre that occupies 4 squares is inside a spell's area, we expect it to take damage once only, not four times. Either it's a target and it's affected like a target, or it's not and isn't.
Where is it written? The exception that proves the rule
There doesn't seem to be a clear rule anywhere that actually says that "partially within the area is still within the area", but it can be seen by when there is a stated exception.
As an example of this, the stinking cloud spell says:
Each creature that is completely within the cloud at the start of its turn must make a Constitution saving throw against poison.
Why does it specify 'completely within the cloud' rather than just saying 'in the cloud'? The reason is because normally being partially in an area counts as being in the area, and this particular effect is an exception where it has to completely engulf a creature to affect it.
Similarly, thunderwave specifies that
unsecured objects that are completely within the area of effect are automatically pushed
...which is only necessary if "objects that are within the area" would normally include things that are even partially in the area.
The exception proves the rule; the existence of an exception is proof that an unstated rule exists.
What are the limits?
So this brings up the question of how big an object is. Is a castle an object? Is a ship an object? If you straighten out a rope, can the whole thing glow if you only catch one end in the area? If a coiled rope is in the area, and then you uncoil it, does it keep glowing?
These are questions a DM will have to grapple with, but in general the DMG says that "an object is a discrete, inanimate item [...], not a building or a vehicle that is composed of many other objects". So a ship or castle should not light up from a single spell.
I see no harm in allowing that a rope lights up if you hit one end of it with faerie fire, but if you as DM find that there's some trick the players are trying to exploit, it's within your realm to declare that it doesn't work that way.