Custom PCB for temperature measurement
It is absolutely possible to fit that into a PCB that size, you can give it a quick try using an online PCB design service such as EasyEDA. What do you mean by 'a custom microcontroller'? If you are willing to do some SMD work, you could get an ATmega328P on the board, and a few passive components to make to work. Then you can either save data to an onboard SD card, or to a connected computer.