Hi,
It depends on what you're using as a reference for the ADC0808. If you use the 5V supply as a reference, you shouldn't have any trouble with clipping assuming your input is positive, i.e. from 0V to 1V. You'll have better resolution though if you also use a voltage reference close to 1V. I think 1.25V is a common value for voltage reference chips, diodes.
If your input is symmetric about 0V, you'll need to bias it to work the ADC. Watch your impedances though since some biomedical sensors have high output impedances.
Regards,
Chris