There are many potential issues with error messaging. Problems such as data type, domain, memory etc. are all internal and misuse issues and not issues of a broken device. If there are no problems with the user input and there are still error messages then it is most likely an issue with the keyboard or the motherboard malfunctioning. This problem would most likely be solved by the replacement of the TI-80 motherboard.
The next most common problem with the power would be the physical button being either broke or stuck. In order to repair the button(s) remove the faceplate of the device and make sure there is nothing interfering the motion of the button(s) then reinstall the faceplate and test the problem button(s). If the faceplate has been removed and the inside of the calculator is clean then the next most common solution would be the replacement of the keyboard.
A common issue with the TI-80 is the power function failing to turn on the calculator. This can include problems with both the battery and/or the power button. The most common cause of the power not initializing would be a dead battery. This is the most simple of fixes, it can be restored with the replacement of both 3-Volt CR2032 Lithium Coin cell batteries.