Which of the following options is strictly less than 0.65?

The correct answer is $0.\overline{64}$.

To find which option is strictly less than 0.65, we can expand any repeating decimals and compare each value to 0.65 place by place (tenths, hundredths, thousandths, etc.).

1. Evaluating $0.\overline{66}$:
The bar over 66 indicates a recurring decimal:
$0.\overline{66}=0.666666...$
Comparing 0.6666... with 0.65, the digit in the hundredths place for 0.6666... is 6, which is greater than 5. Therefore, 0.6666... > 0.65.

2. Evaluating $0.\overline{64}$:
The bar over 64 indicates a recurring decimal:
$0.\overline{64}=0.646464...$
Comparing 0.6464... with 0.65, the digit in the hundredths place for 0.6464... is 4, which is strictly less than 5. Therefore, 0.6464... < 0.65.

3. Evaluating 0.67:
Comparing 0.67 with 0.65, the hundredths digit is 7, which is greater than 5. Therefore, 0.67 > 0.65.

4. Evaluating 0.70:
Comparing 0.70 with 0.65, the tenths digit is 7, which is greater than 6. Therefore, 0.70 > 0.65.

Thus, the only value that is strictly less than 0.65 is $0.\overline{64}$.