Hey there. Sorry to read about your situation, I hope I can help you a little bit since I'm German.
Sabbaticals are not uncommon in Germany any more, but are at will of your employer. Essentially it is just an unpaid leave and a promise that you'll still have your job.
Small companies, like your employer, which is a startup, will probably not have enough people to easily compensate while your gone and have a harder time to find a replacement for just the time in which your away.
Especially because you working overtime already, they seem to have a too much work for too less people.
A Freelancer would be easier to get, but they are much more expensive than regular employees.
But in general it depends on the company, whether they see employees are resources or as the engine running the company. If you talk to your boss, that might give you an idea whether unpaid leave is an option. And he might also recognize that the workload is too high.
There are no legal provisions for sabbaticals. Also the job center will not immediately pay an unemployment benefit if you choose to quit for your personal reasons. At least you'll not get any pay for the first three (?) months. If and what pay you'll receive afterwards depends on if you are eligible and how much contribution you have paid in the past.
There is another option: Talk to a doctor. Potential burn out is a serious topic and any good doctor should recognize the signs and prevent anything worse. You're health care provider might also be able to help you there, most likely they sponsor yoga sessions. But calling their help line for medical advise is certainly worth it. Doctors might be able to put you on sick leave, but most likey just a few days until you have serious symptoms.
A side note:
Be aware that any treatment you receive might affect your rating for occupational disablement insurances. If you don't have one (which you should as long as you're healthy!), and you choose to get one any time within the next five years, then they'll request all diagnosis from all doctors you have been seeing. Any illness / reason for which you have been seeing a doctor will most likely be excluded from the insurance.
Some psychotherapists unfortunately tend to write stuff (some standard clauses) into your file which is not true (ie. anxiety disorder because you can't sleep), but makes them get more money from the health care provider. Thus if you don't have an occupational disablement insurance and choose to visit a psychotherapist, make sure you know what diagnosis they are noting down.
Hope this helps! Best of luck!