First of all, sorry for the paint, but I tried my best. This is my first project, so I wanted to see if everything was connected properly and if anyone had any tips or suggestions. Goal is to use buttons to navigate through pages of text displayed on the screen. Due to space limitations, I'm using 6 x SR936SW batteries (if you know of a circular battery that's 10-11mm in diameter, feel free to let me know)
Well, first of all, why don't you use the internal pull-ups on the ESP8266 for your buttons instead of using the external pull-up resistors? That will save you a lot of space in your design. If your OLED doesn't have pullups though you'll want to have 10kOhm resistors on your SDA and SCL lines. Also, 3-3.1V is not the optimal operating voltage of the ESP8266 chip. That is 3.3V. Your batteries only have an optimal output of 140uA though. Your ESP8266 will use around 300 mA. So, while your battery capacity in theory would get you about 20 minutes of runtime, the batteries are not designed to output that level of current. I would recommend using a LiPo battery instead. There are small version (~150 mAh) that would work much better. Also, your ESP-01S isn't going to be very tolerant of the voltage variance from a battery. You need a board with built-in battery circuitry like an Adafruit Feather board.
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Due to space limitations, this project requires everything to fit within a square of 12,2 mm or a circle of 10 mm. Therefore, I use the ESP01S, which can fit diagonally in a 12,2mm square if you take off the pins.
The internal buttons aren't where I need them to be, so that's why.
What are OLED pullups and how to check if mine has it?
I do not have a length specification, but everything must be shorter than 140-150mm. Putting everything in a line is
25(esp) + 38(oled) + 15(2xbuttons) + 20(some free space) = ~100mm
In that case, I need at least 40-50mm for the battery, and if I use LiPo, I also need a 3.7v-3.3v regulator. I couldn't find a battery and regulator that small. It would also be nice to have a connector on the battery so I can disconnect and recharge it, which would also require some space.
(What are uA used for)? I don't understand this part, can you please explain it to me? By turning off wifi, esp01s will use 30mAh (I saw that somewhere, but I'm not sure if that's true), and oleds use about the same amount. If I have 150 mah and use 60 mah, doesn't that make 3 hours of run time? It only needs to run for 2-3 hours at a time.
mA is the is milliamps which is the current draw. Even if you were right and the module only pulled 30 mA, that is still over 200x as high as the normal draw on these batteries, 140 uA (microamps). They are not going to power your device. Those batteries are not high current output batteries. They are designed to give low amounts of power over a long time. You need a battery with a high output current.
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What if I used SR936W? They are same as SR936SW but high drain?
I think you need to do a prototype and measure your "on" time in a real environment... INCLUDING cycling power On/Off so batteries can relax some. *
- Chemistry determines the voltage and mass of active materials determine total mAH ... but, cycling permits gases and temp to stabilize over off-time. At 30mA likely not relevant bit I mention it out of completeness.
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If it's a display, you can't really turn it off, can you?
I'm using one of the buttons to clear display when I don't use it. So I think that saves some power
No, that will not help. What @mrburnette meant was that if you sleep the board after only a few seconds of being on (like the time it takes to take a few sensor readings) that will give the batteries time to rest as well. You are going to try and pull too much current from these batteries and it will not work.
Sorry don't get it 
I'm newbie here. Would you mind explaining it to me in simple terms?
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