I am trying to find a simple hardware solution to implement a 2-bit timer that gives out 2-bits continuously repeating data 00, 01, 10, 11, 00, 01.... whose time period is approx 1ms. The accuracy and jitter in the period is not an important design parameter. The simplicity in design is important. Simplicity means without any programmable logic. The voltage levels can be any.
Use a 74xx4060 14-bit binary counter IC, such as the 5 V 74HCT4060. This is a cheap and readily available part and uses a resistor and a capacitor to control the increment rate of a 14-bit counter.
Its 10 output counter bits are different stages of a 14-bit ripple counter. So you can choose a relatively high oscillator frequency that requires a small capacitor and high resistor, then use a divider stage that gives you your 1 kHz 2-bit value.
It uses an external resistor and capacitor to set its internal oscillator frequency.
If the accuracy of an RC oscillator is insufficient, you can use a crystal instead.