It's really dependent on where you live, but in many places, hospital staff aren't respectful or kind to people who have self-harmed. It's unlikely you'd get a bunch of time to really rest in a standard hospital either; ERs are loud and busy, nurses and staff check on you and ask you questions frequently. With something like cuts, those can be addressed medically quite fast, then they'll need to discharge you to have room for other people.
Going to an inpatient psychiatric unit means you could have days worth of time, but I'm sure you're aware of the downsides of that since you say you don't want to go there. Some people do actually find it a "break" going to an inpatient unit, but it's very dependent on what the units in your area are like and what your individual needs are.
I think there are so many options that are better to try if you simply want time to rest. I don't know what is preventing you from resting at home, but I'm assuming you mean something like work or college classes where you're expected to show up daily. If you're considering physically hurting yourself to get out of going, to me that's "sick" enough to justify not going; you clearly need a rest or break of a some kind, so calling in sick is completely justified IMO.
And then if you're seeking support or comfort, I would by far recommend looking for a caring, respectful therapist who you can see-long term. I know that's a process to find one and not an immediate fix, but if you find a good one you'll have someone who you can reliably go to for breaks from mental pressure from then on.
And besides that, SaSu is a great place to vent about whatever you're going through and have people who've been there and don't judge. I just wouldn't recommend self-harming with the plan of going to a hospital (when you don't want to be admitted to inpatient psych), because it's such a gamble and you don't want things to get worse when you already feel pretty shitty.
