How many digits?

Answer: 35

Using estimation

$38 ~659 ...\approx 4\times10^{19}$ ($=4000...0$ (19 zeroes))

$342 ~973 ... \approx 3\times10^{14}$ ($=300...0$ (14 zeroes))

$4\times10^{19}\times 3\times10^{14}=12\times10^{33}$, which has 33 zeroes so 35 digits.

But the first number was rounded up, the second number was rounded down. We need to round both up or both down.

$342 ... \approx 4\times10^{14}$ (rounded up)

$4\times10^{19}\times 4\times10^{14}=16\times10^{33}$ which also has 35 digits.

$38 ...\approx 3.8\times10^{19}$ (rounded down)

$3.8\times10^{19}\times 3\times10^{14}=11.4\times10^{33}$ which also has 35 digits.

Comparing to an easier product

$38 ~659 ...=3.865...\times10^{19}$

$342 ~973 ... =3.429...\times10^{14}$

Product $=3.865...\times10^{19}\times 3.429...\times10^{14}$ has 34 digits if $3.865...\times 3.429...\lt10$, and 35 digits if it is $\gt10$

$3.333...\times3=10$ and $3.333...\lt3.8...$, $3\lt3.4...$ so $3.865...\times 3.429...\gt10$