RAW, the cat cannot end its turn in the same space as you
In the PHB (p. 191), the rules for Moving Around Other Creatures states:
Whether a creature is a friend or an enemy, you can't willingly end your move in its space.
Since the cat cannot end its move - and therefore its turn - in your space, it also cannot hide in your space from the enemy. If it could, then it wouldn't trigger opportunity attacks, since that's triggered by leaving the enemy's reach (this is assuming that you are within the enemy's reach).
As it stands, it would need to leave your space to hide, and if that means leaving the enemy's reach, it would then take an opportunity attack unless it disengaged first (although it then would have used its action, so cannot hide until its next turn).
However, given that the cat familiar is a Tiny, non-combative (I assume? Familiars, at least those via the find familiar spell, are usually non-combative, unless you're a Pact of the Chain Warlock) creature, then a DM may overlook this and let the cat hide in your space, but that's up to the DM.
The cat can mount you though
The one exception to this (thanks to @Darth Pseudonym for pointing this out) is that, since the cat is Tiny and you are Small, then if the cat were to use half its movement to mount you, it could then share your space. Since a humanoid can be a mount for another humanoid, I don't see why this wouldn't work.
From the Mounted Combat rules:
A willing creature that is at least one size larger than you and that has an appropriate anatomy can serve as a mount, using the following rules.
Once during your move, you can mount a creature that is within 5 feet of you or dismount. Doing so costs an amount of movement equal to half your speed.
Being mounted by your cat shouldn't impose any restrictions on you whatsoever, since you would be considered an independent mount:
An independent mount retains its place in the initiative order. Bearing a rider puts no restrictions on the actions the mount can take, and it moves and acts as it wishes.
I don't think the cat can hide while mounted, but at the same time I don't know if there's anything that says that someone who is mounted can't hide either. The closest thing I can find is the Hiding sidebar, which says:
You can't hide from a creature that can see you clearly...
The if the cat can't hide, the cat would still be able to be attacked, since:
... if the mount provokes an opportunity attack while you’re on it, the attacker can target you or the mount.