Your reasons for not trying seem to be based on what you think might be the case. The reality is, busy people who work at busy companies get new jobs all the time. You need to actually schedule an interview before you'll know whether any of your "reasons" apply to that specific company.
My company doesn't have vacation days or sick days - when someone goes on vacation, they just sort of do.
Then just take a day off like everyone else does. Notify your team in the usual ways that the team notifies each other. Job interviews rarely come as a surprise, so you should be able to plan at least a few days ahead.
But wait... You're probably thinking "But what if they ask why I need the day off?" Try this...
We've all been working long hours and I need to see to some things at home before they become problems.
In case they press you further, have a list of things that you intend to do that day, so you can be honest. You don't have to tell them everything you plan to do that day.
Ok, that works for one day, but what about having to take more than one day here and there?
Well, in this case, if you can't reasonably take a day here and there, or a half-day, etc., then you really should take a little time away for lunch, and maybe go out and network with friends for an hour. That might make the job hunt a little easier, and you might be able to get an inside track on another job through a friend, where the interview process can be expedited.
Now for the rest of your concerns...
...this is the tech industry, so interviews last for at least a few hours, if not all day. Google interviews, for example, take an entire day.
Not true. It depends on the company holding the interview. Don't sit on the sidelines just because you think the interview process will require long hours. Schedule the interview and found out for sure what the time commitment will be.
Any company that would have staff interviewing someone at that time (so, working at that time) would be one that I'd want to rule out on my end anyways.
Maybe. Maybe not. Maybe the company understands that there are good people working in tough situations and are willing to be accommodating. Again, you won't know until you try. Don't shut down an opportunity because of unfounded preconceived notions.
"Pretend to have a doctor's appointment"
You're right to be uncomfortable with this option. Be honest. It really is the best way to go. You'll never have to worry about unraveling a lie (or a lie unraveling itself, which they seem to always do, eventually) if you just don't tell lies.