ULN2803A & 74HC595 Wiring

At 250mA, and a typical volt drop of ~1.2volt across the switch transistor, there is 1.2*0.25= 0.3watt per channel generated inside that ULN chip. With eight channels 'on', that's 2.4watt...
That will push the temp of silicon inside the IC up with about 175C (see datasheet).
When the chip cracks under that stress, it could well take out the drive chip, and that could take out the Arduino.
Leo..