So how can I look for a job? I'll have to answer phone calls during working hours from my current work place, and I will have to go to job interviews. Is this even possible without quitting or letting the employer know?
Absolutely I'd want to keep them, but sometimes that's not possible. Even if they leave, I'd help them remain productive, plan their exit, and control the hand-over.
If finding another job is the path they'd take, I'd find a way to let people take calls and attend interviews. Obviously we'd discuss either longer hours to make up the time, or a reduction in pay, but we'd find a way to agree how it would work.
If you can't talk to your current boss, moving out is the best thing to do, but the least you can do it talk to them first and give them the chance to make things better, or help you transition to a job where you're happier.
As for calls and interviews, this is what voicemail is for. Call back on your breaks. People understand the situation. Also interviews can usually be done before or after work. If the new employer is not understanding it might be a sign they're not a great company to work for.
This is a little dishonest, but why not tell your employer you have a doctor's appointment or jury duty? Or just say you'll be out of office for personal reasons. Or take a day off and interview one half of the day and use the rest to relax.
You can always schedule job interviews on Saturdays. Again, most interviewers understand the circumstances and usually do hiring on Saturdays to accommodate those who are working.
Hopefully, taking a vacation day (or half day) isn't impossible at your current job. Having appointments in the middle of the day isn't unusual. Make sure all phone calls are scheduled and step out to take them.