My $20 "Pittsburgh" calipers from Harbor Freight (don't judge) are still on their original batteries, after 6 years. They have a power button, and a separate zero button.
They don't actually turn off though. It just turns the display off. They need to continuously sense position since digital calipers measure movement increments rather than absolute position. That's why if you take the battery out and put it back in it will prompt you to re-zero the calipers.
