The spell magic jar puts your soul into a jar, and your body is in a catatonic state.
What are these both treated as? Is the jar counted as an object, a construct, or a creature in general? Is your catatonic body counted as a corpse (therefore an object) an asleep creature, or something else?
I ask this because there are quite a few spells that say you must target a creature with it, so I wondered if either of these counted as such.
The jar is an object, matching the object definition on p. 204 PHB.
For the purpose of these rules, an object is a discrete, inanimate item like a window, door, sword, book, table, chair, or stone
The jar itself is not sentient, it is just housing the soul, and that does not make it a creature:
While your soul inhabits the container, you are aware of your surroundings as if you were in the container's space.
The body is a creature. In general, a creature is a body with a soul although there can be exceptions (see for example, many inhabitants of Barovia in Curse of Strahd); your soul is still your soul, just out of body.
The body is described as catatonic which means "in an immobile or unresponsive stupor" and alive, and only creatures can be alive. It is not inanimate (i.e. "lacking life"), as an object needs to be. The text "if your body is dead when you return to it...", also demonstrates that the body normally is not dead, thanks Mindwin.)
But this is an unusual setup, and neither the spell nor the rules do explicitly really say as what the body or container does count, so I'd expect the DM will have to decide how to handle this.
The material components reads
a gem, crystal, reliquary, or some other ornamental container worth at least 500 gp.
The example provided are all objects, and the spell description does not make any specific rule modifying this aspect. Therefore, any spell that targets creatures can not target the jar.
There are some exceptions, though, that may require some ruling from the DM: for example, if the caster uses a decorated wooden crate (worthing 500 gp) and it happens that such crate is inside the AoE of a Fireball it can be damaged.
There are no description in game terms what happen to the body (as in the case of Feign Death spell: the target appears dead but they are still a creature under blinded and incapacitated conditions), but the spell's text says (emphasis mine):
[...] if your body is dead when you attempt to return to it, you die.
Since only creatures can die\$^1\$, this means that the body is a creature.
\$^1\$ Credits to Mindwin for having pointed this out.
Honestly, catatonic bodies without souls and jars with souls are far enough from the regular cases of "object" and "creature" that this is something that should be answered by your DM on a case-by-case basis depending on the effect.
And if your PC wants to know the answer, they should either be doing skill checks or experiments, because this isn't going to be a freebie either.
Any general answer would or could lead to strange results. Like, if you charm a catatonic body without a soul, is the creature the soul is currently possessing effected by the charm? If you charmed them before they cast magic jar, does it follow the soul, or stay with the body?
When running into corner cases in 5e D&D with its explicit emphasis on "rulings not rules", you shouldn't avoid seeking a general rule. As a DM, you should feel free to rule the body a creature for one effect, and not for another, and the same for the ensouled jar; and, as player controlling a PC, you should understand that things may not work the way you expect, and you should have no expectation of knowing how they will work before you try it or ask the DM.
