Now I have done a number of experiments with powering the ESP8266-01 module with batteries. Changed software, voltage converters and deep sleep modes.
For the voltage converters I used the LM1117-3V3, a step-down converter, and the MCP1702-3302E. The conclusion is that the MCP1702-3302E is the best solution because it has a minimal quiescent current with no load.
3x AAA seems to be the best solution for the batteries:
For the software it was possible to use a maximum of 4 seconds per measurement (power up) and then go back to deep sleep modes.
You have to make some modification to the ESP8266-01 module to get as less possible current when in deep sleep.
On the left a connection beween two GPIO pins to let the timer reset the chip that is in deep sleep. On the right the power led is removed to reduces current in deep sleep mode.